##### Class 8^{th} Mathematics AP Board Solution

**Exercise 6.1**- 39 What will be the units digit of the square of the following numbers?…
- 297 What will be the units digit of the square of the following numbers?…
- 5125 What will be the units digit of the square of the following numbers?…
- 7286 What will be the units digit of the square of the following numbers?…
- 8742 What will be the units digit of the square of the following numbers?…
- 121 Which of the following numbers are perfect squares?
- 136 Which of the following numbers are perfect squares?
- 256 Which of the following numbers are perfect squares?
- 321 Which of the following numbers are perfect squares?
- 600 Which of the following numbers are perfect squares?
- 257 The following numbers are not perfect squares. Give reasons?
- 4592 The following numbers are not perfect squares. Give reasons?…
- 2433 The following numbers are not perfect squares. Give reasons?…
- 5050 The following numbers are not perfect squares. Give reasons?…
- 6098 The following numbers are not perfect squares. Give reasons?…
- 431 Find whether the square of the following numbers are even or odd?…
- 2826 Find whether the square of the following numbers are even or odd?…
- 8204 Find whether the square of the following numbers are even or odd?…
- 17779 Find whether the square of the following numbers are even or odd?…
- 99998 Find whether the square of the following numbers are even or odd?…
- 25; 26 How many numbers lie between the square of the following numbers…
- 56; 57 How many numbers lie between the square of the following numbers…
- 107;108 How many numbers lie between the square of the following numbers…
- 1 + 3 + 5 + 7 + 9 = Without adding, find the sum of the following numbers…
- 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = Without adding, find the sum of the…
- 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 = Without adding,…

**Exercise 6.2**- 441 Find the square roots of the following numbers by Prime factorization…
- 784 Find the square roots of the following numbers by Prime factorization…
- 4096 Find the square roots of the following numbers by Prime factorization…
- 7056 Find the square roots of the following numbers by Prime factorization…
- Find the smallest number by which 3645 must be multiplied to get a perfect…
- Find the smallest number by which 2400 is to be multiplied to get a perfect…
- Find the smallest number by which 7776 is to be divided to get a perfect square.…
- 1521 trees are planted in a garden in such a way that there are as many trees in…
- A school collected ` 2601 as fees from its students. If fee paid by each student…
- The product of two numbers is 1296. If one number is 16 times the other, find…
- 7921 soldiers sat in an auditorium in such a way that there are as many soldiers…
- The area of a square field is 5184 m^2 . Find the area of a rectangular field,…

**Exercise 6.3**- 1089 Find the square roots of the following numbers by division method.…
- 2304 Find the square roots of the following numbers by division method.…
- 7744 Find the square roots of the following numbers by division method.…
- 6084 Find the square roots of the following numbers by division method.…
- 9025 Find the square roots of the following numbers by division method.…
- 2.56 Find the square roots of the following decimal numbers.
- 18.49 Find the square roots of the following decimal numbers.
- 68.89 Find the square roots of the following decimal numbers.
- 84.64 Find the square roots of the following decimal numbers.
- Find the least number that is to be subtracted from 4000 to make it perfect…
- Find the length of the side of a square whose area is 4489 sq.cm.…
- A gardener wishes to plant 8289 plants in the form of a square and found that…
- Find the least perfect square with four digits.
- Find the least number which must be added to 6412 to make it a perfect square?…
- Estimate the value of the following numbers to the nearest whole number…
- root 250 Estimate the value of the following numbers to the nearest whole…
- root 780 Estimate the value of the following numbers to the nearest whole…

**Exercise 6.4**- 8 Find the cubes of the following numbers
- 16 Find the cubes of the following numbers
- 21 Find the cubes of the following numbers
- 30 Find the cubes of the following numbers
- 243 Test whether the given numbers are perfect cubes or not.
- 516 Test whether the given numbers are perfect cubes or not.
- 729 Test whether the given numbers are perfect cubes or not.
- 8000 Test whether the given numbers are perfect cubes or not.
- 2700 Test whether the given numbers are perfect cubes or not.
- Find the smallest number by which 8788 must be multiplied to obtain a perfect…
- What smallest number should 7803 be multiplied with so that the product becomes…
- Find the smallest number by which 8640 must be divided so that the quotient is a…
- Ravi made a cuboid of plasticine of dimensions 12cm, 8cm and 3cm. How many…
- Find the smallest prime number dividing the sum 3^11 + 5^13 .

**Exercise 6.5**- 343 Find the cube root of the following numbers by prime factorization method.…
- 729 Find the cube root of the following numbers by prime factorization method.…
- 1331 Find the cube root of the following numbers by prime factorization method.…
- 2744 Find the cube root of the following numbers by prime factorization method.…
- 512 Find the cube root of the following numbers through estimation?…
- 2197 Find the cube root of the following numbers through estimation?…
- 3375 Find the cube root of the following numbers through estimation?…
- 5832 Find the cube root of the following numbers through estimation?…
- Cube of an even number is an odd number State true or false?
- A perfect cube may end with two zeros State true or false?
- If a number ends with 5, then its cube ends with 5 State true or false?…
- Cube of a number ending with zero has three zeros at its right State true or…
- The cube of a single digit number may be a single digit number. State true or…
- There is no perfect cube which ends with 8 State true or false?
- The cube of a two-digit number may be a three-digit number. State true or…
- Find the two-digit number which a square number is and also a cubic…

**Exercise 6.1**

- 39 What will be the units digit of the square of the following numbers?…
- 297 What will be the units digit of the square of the following numbers?…
- 5125 What will be the units digit of the square of the following numbers?…
- 7286 What will be the units digit of the square of the following numbers?…
- 8742 What will be the units digit of the square of the following numbers?…
- 121 Which of the following numbers are perfect squares?
- 136 Which of the following numbers are perfect squares?
- 256 Which of the following numbers are perfect squares?
- 321 Which of the following numbers are perfect squares?
- 600 Which of the following numbers are perfect squares?
- 257 The following numbers are not perfect squares. Give reasons?
- 4592 The following numbers are not perfect squares. Give reasons?…
- 2433 The following numbers are not perfect squares. Give reasons?…
- 5050 The following numbers are not perfect squares. Give reasons?…
- 6098 The following numbers are not perfect squares. Give reasons?…
- 431 Find whether the square of the following numbers are even or odd?…
- 2826 Find whether the square of the following numbers are even or odd?…
- 8204 Find whether the square of the following numbers are even or odd?…
- 17779 Find whether the square of the following numbers are even or odd?…
- 99998 Find whether the square of the following numbers are even or odd?…
- 25; 26 How many numbers lie between the square of the following numbers…
- 56; 57 How many numbers lie between the square of the following numbers…
- 107;108 How many numbers lie between the square of the following numbers…
- 1 + 3 + 5 + 7 + 9 = Without adding, find the sum of the following numbers…
- 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = Without adding, find the sum of the…
- 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 = Without adding,…

**Exercise 6.2**

- 441 Find the square roots of the following numbers by Prime factorization…
- 784 Find the square roots of the following numbers by Prime factorization…
- 4096 Find the square roots of the following numbers by Prime factorization…
- 7056 Find the square roots of the following numbers by Prime factorization…
- Find the smallest number by which 3645 must be multiplied to get a perfect…
- Find the smallest number by which 2400 is to be multiplied to get a perfect…
- Find the smallest number by which 7776 is to be divided to get a perfect square.…
- 1521 trees are planted in a garden in such a way that there are as many trees in…
- A school collected ` 2601 as fees from its students. If fee paid by each student…
- The product of two numbers is 1296. If one number is 16 times the other, find…
- 7921 soldiers sat in an auditorium in such a way that there are as many soldiers…
- The area of a square field is 5184 m^2 . Find the area of a rectangular field,…

**Exercise 6.3**

- 1089 Find the square roots of the following numbers by division method.…
- 2304 Find the square roots of the following numbers by division method.…
- 7744 Find the square roots of the following numbers by division method.…
- 6084 Find the square roots of the following numbers by division method.…
- 9025 Find the square roots of the following numbers by division method.…
- 2.56 Find the square roots of the following decimal numbers.
- 18.49 Find the square roots of the following decimal numbers.
- 68.89 Find the square roots of the following decimal numbers.
- 84.64 Find the square roots of the following decimal numbers.
- Find the least number that is to be subtracted from 4000 to make it perfect…
- Find the length of the side of a square whose area is 4489 sq.cm.…
- A gardener wishes to plant 8289 plants in the form of a square and found that…
- Find the least perfect square with four digits.
- Find the least number which must be added to 6412 to make it a perfect square?…
- Estimate the value of the following numbers to the nearest whole number…
- root 250 Estimate the value of the following numbers to the nearest whole…
- root 780 Estimate the value of the following numbers to the nearest whole…

**Exercise 6.4**

- 8 Find the cubes of the following numbers
- 16 Find the cubes of the following numbers
- 21 Find the cubes of the following numbers
- 30 Find the cubes of the following numbers
- 243 Test whether the given numbers are perfect cubes or not.
- 516 Test whether the given numbers are perfect cubes or not.
- 729 Test whether the given numbers are perfect cubes or not.
- 8000 Test whether the given numbers are perfect cubes or not.
- 2700 Test whether the given numbers are perfect cubes or not.
- Find the smallest number by which 8788 must be multiplied to obtain a perfect…
- What smallest number should 7803 be multiplied with so that the product becomes…
- Find the smallest number by which 8640 must be divided so that the quotient is a…
- Ravi made a cuboid of plasticine of dimensions 12cm, 8cm and 3cm. How many…
- Find the smallest prime number dividing the sum 3^11 + 5^13 .

**Exercise 6.5**

- 343 Find the cube root of the following numbers by prime factorization method.…
- 729 Find the cube root of the following numbers by prime factorization method.…
- 1331 Find the cube root of the following numbers by prime factorization method.…
- 2744 Find the cube root of the following numbers by prime factorization method.…
- 512 Find the cube root of the following numbers through estimation?…
- 2197 Find the cube root of the following numbers through estimation?…
- 3375 Find the cube root of the following numbers through estimation?…
- 5832 Find the cube root of the following numbers through estimation?…
- Cube of an even number is an odd number State true or false?
- A perfect cube may end with two zeros State true or false?
- If a number ends with 5, then its cube ends with 5 State true or false?…
- Cube of a number ending with zero has three zeros at its right State true or…
- The cube of a single digit number may be a single digit number. State true or…
- There is no perfect cube which ends with 8 State true or false?
- The cube of a two-digit number may be a three-digit number. State true or…
- Find the two-digit number which a square number is and also a cubic…

###### Exercise 6.1

**Question 1.**What will be the units digit of the square of the following numbers?

39

**Answer:**Given a number 39 and need to find out the unit digit of the square of number.

Now, the square of number, n = n^{2}

Since, we have “9” as the unit place in the number “39”.

⇒ 9^{2} = 81

⇒ 1 will be the unit digit of square of 39.

Hence, the units’ digit of the square of 39 is 1.

**Question 2.**What will be the units digit of the square of the following numbers?

297

**Answer:**Given a number 297 and need to find out the unit digit of the square of number.

Now, the square of number, n = n^{2}

Since, we have “7” in the units place in the given number “297” we have to square the number 7

⇒ 7^{2} = 49 and 9 will be the units’ digit.

Hence, the units digit of the square of 297 is 9.

**Question 3.**What will be the units digit of the square of the following numbers?

5125

**Answer:**Given a number 5125 and need to find out the unit digit of the square of number.

Now, the square of number, n = n^{2}

Since, we have “5” in the units place in the given number “5125” we have to square the number 5

⇒ 5^{2} = 25 and 5 will be the units’ digit.

Hence, the units digit of the square of 5125 is 5.

**Question 4.**What will be the units digit of the square of the following numbers?

7286

**Answer:**Given a number 7286 and need to find out the unit digit of the square of number.

Now, the square of number, n = n^{2}

Since, we have “6” in the units place in the given number “7286” we have to square the number 6

⇒ 6^{2} = 36

⇒ 6 will be the units’ digit.

Hence, the units digit of the square of 7286 is 6.

**Question 5.**What will be the units digit of the square of the following numbers?

8742

**Answer:**Given a number 8742 and need to find out the unit digit of the square of number.

Now, the square of number, n = n^{2}

Since, we have “2” in the units place in the given number “8742” we have to square the number 2

⇒ 2^{2} = 4

⇒ 4will be the units’ digit.

Hence, the units digit of the square of 8742 is 4.

**Question 6.**Which of the following numbers are perfect squares?

121

**Answer:**Given, a number 121 and need to find out whether it is perfect square or not.

Now, we have the perfect square = , [T1] where m and n are integers

⇒ 121 is a perfect square as it can be expressed as 11 × 11 form the product of two equal integer.

⇒121 = 11 × [T2] 11

Hence, 121 is a perfect square.

**Question 7.**Which of the following numbers are perfect squares?

136

**Answer:**Given a number 136 and need to find out whether it is perfect square or not.

Now, we have the perfect square = , where m and n are integers

⇒ 136 is not a perfect square as it cannot be expressed as n × n form the product of two equal integer.

Hence, 136 is not a perfect square.

**Question 8.**Which of the following numbers are perfect squares?

256

**Answer:**Given a number 256 and need to find out whether it is perfect square or not.

Now, we have the perfect square = , where m and n are integers

⇒ 256 is a perfect square as it can be

expressed as 16 × 16 form the product of two equal integer.

⇒ 256 = 16 × 16

Hence, 256 is a perfect square.

**Question 9.**Which of the following numbers are perfect squares?

321

**Answer:**Given a number 321 and need to find out whether it is perfect square or not.

Now, we have the perfect square = , where m and n are integers

⇒ 321 is not a perfect square as it cannot be

expressed as n × n form the product of two equal integer.

Hence, 321 is not a perfect square.

**Question 10.**Which of the following numbers are perfect squares?

600

**Answer:**Given a number 600 and need to find out whether it is perfect square or not.

Now, we have the perfect square = , where m and n are integers

⇒ 600 is not a perfect square as it cannot be expressed as n × n form the product of two equal integer.

Hence, 600 is not a perfect square.

**Question 11.**The following numbers are not perfect squares. Give reasons?

257

**Answer:**Given a number 257 is not perfect square. Need to find out the reason.

Now, we have the perfect square = , where m and n are integers

⇒257 is not a perfect square as it cannot be expressed as n × n form the product of two equal

integer and the number that have 2,3,7 or 8 in the units place are not perfect squares.

Hence, the given number 257 is not a perfect square.

**Question 12.**The following numbers are not perfect squares. Give reasons?

4592

**Answer:**Given a number 4592 is not perfect square. Need to find out the reason.

Now, we have the perfect square = , where m and n are integers

⇒ 4592 is not a perfect square as it cannot be expressed as n × n form the product of two equal integer and the number that have 2,3,7 or 8 in the units place are not perfect squares.

Hence, the given number 4592 is not a perfect square.

**Question 13.**The following numbers are not perfect squares. Give reasons?

2433

**Answer:**Given a number 2433 is not perfect square. Need to find out the reason.

Now, we have the perfect square = , where m and n are integers

⇒ 2433 is not a perfect square as it cannot be expressed as n × n form the product of two equal integer and the number that have 2,3,7 or 8 in the units place are not perfect squares.

Hence, the given number 2433 is not a perfect square.

**Question 14.**The following numbers are not perfect squares. Give reasons?

5050

**Answer:**Given a number 5050 is not perfect square. Need to find out the reason.

Now, we have the perfect square = , where m and n are integers

⇒ 5050 is not a perfect square as it cannot be expressed as n × n form the product of two equal integer.

Hence, the given number 5050 is not a perfect square.

**Question 15.**The following numbers are not perfect squares. Give reasons?

6098

**Answer:**Given a number 6098 is not perfect square. Need to find out the reason.

Now, we have the perfect square = , where m and n are integers

⇒ 6098 is not a perfect square as it cannot be expressed as n × n form the product of two equal integer and the number that have 2,3,7 or 8 in the units place are not perfect squares.

Hence, the given number 6098 is not a perfect square.

**Question 16.**Find whether the square of the following numbers are even or odd?

431

**Answer:**Given a number 431 and need to find out whether it is even or odd

Now, the square of number, n = n^{2}

Consider units digit of a number 431 and square the units digit number.

⇒ Here, Units digit number is 1

⇒ Square of 1 = 1

⇒ Square the unit digit 1 = 1

⇒ 1 is a odd number

⇒ ∴ square of 431 will be again odd number

Hence, 431 is odd number.

**Question 17.**Find whether the square of the following numbers are even or odd?

2826

**Answer:**Given a number 2826 and need to find out whether it is even or odd

Now, the square of number, n = n^{2}

Consider units digit of a number 2826 and square the units digit number.

⇒ Here, Units digit number is 6

⇒ Square of 6 = 36

⇒ 6 is a even number

⇒ ∴ square of 2826 will be again even number

Hence, 2826 is even number.

**Question 18.**Find whether the square of the following numbers are even or odd?

8204

**Answer:**__Given:__ A number 8204 and need to find out whether it is even or odd

Now, the square of number, n = n^{2}

Consider units digit of a number 8204 and square the units digit number.

⇒ Here, Units digit number is 4

⇒ Square of 4 = 16

⇒ 4 is a even number

⇒ ∴ square of 8204 will be again even number

Hence, 8204 is even number.

**Question 19.**Find whether the square of the following numbers are even or odd?

17779

**Answer:**Given a number 17779 and need to find out whether it is even or odd

Now, the square of number, n = n^{2}

Consider units digit of a number 17779 and square

the units digit number.

⇒ Here, Units digit number is 9

⇒ Square of 9 = 81

⇒ 9 is a odd number

⇒ ∴ square of 17779 will be again odd number

Hence, 17779 is odd number.

**Question 20.**Find whether the square of the following numbers are even or odd?

99998

**Answer:**Given a number 99998 and need to find out whether it is even or odd

Now, the square of number, n = n^{2}

Consider units digit of a number 99998 and square the units digit number.

⇒ Here, Units digit number is 8

⇒ Square of 8 = 64

⇒ 8 is a even number

⇒ ∴ square of 99998 will be again even number

Hence, 99998 is even number.

**Question 21.**How many numbers lie between the square of the following numbers

25; 26

**Answer:**Given, two numbers 25 and 26 we need to find out the numbers lie between the squares of the given numbers.

Now, we have numbers lie between the square of n and (n + 1) as 2n i.e 2 × base of first number.

⇒ Numbers between squares of 25 and (25 + 1) = 2 × 25

⇒ 50

Hence, 50 numbers lies between the square of the given numbers.

**Question 22.**How many numbers lie between the square of the following numbers

56; 57

**Answer:**Given, two numbers 56 and 57 we need to find out the numbers lie between the squares of the given numbers.

Now, we have numbers lie between the square of n and (n + 1) as 2n i.e 2 × base of first number.

⇒ Numbers between squares of 56 and (56 + 1) = 2 × 56

⇒112

Hence, 112 numbers lies between the square of the given numbers.

**Question 23.**How many numbers lie between the square of the following numbers

107;108

**Answer:**Given, two numbers 107 and 108 we need to find out the numbers lie between the squares of the given numbers.

Now, we have numbers lie between the square of n and (n + 1) as 2n i.e 2 × base of first number.

⇒ Numbers between squares of 107 and (107 + 1) = 2 × 107

⇒ 214

Hence, 214 numbers lies between the square of the given numbers.

**Question 24.**Without adding, find the sum of the following numbers

1 + 3 + 5 + 7 + 9 =

**Answer:**Given, 5 consecutive odd numbers sum. To find out, the sum of consecutive odd numbers from “1” to “9” without adding.

We have sum of first n odd numbers = n^{2}.

Here, n = 5

⇒ 1 + 3 + 5 + 7 + 9 = 5^{2}

⇒ 1 + 3 + 5 + 7 + 9 = 25

Hence, the sum of 1 + 3 + 5 + 7 + 9 without adding = 25

**Question 25.**Without adding, find the sum of the following numbers

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 =

**Answer:**Given, 9 consecutive odd numbers sum. To find out, the sum of consecutive odd numbers from “1” to “17” without adding.

We have sum of first n odd numbers = n^{2}.

Here, n = 9

⇒ 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 9^{2}

⇒ 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81

Hence, the sum of 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 without adding = 81

**Question 26.**Without adding, find the sum of the following numbers

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 =

**Answer:**Given, 13 consecutive odd numbers sum. To find out, the sum of consecutive odd numbers from “1” to “25” without adding.

We have sum of first n odd numbers = n^{2}.

Here, n = 13

⇒ 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 = 13^{2}

⇒ 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 = 169

Hence, the sum of 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 without adding = 169

**Question 1.**

What will be the units digit of the square of the following numbers?

39

**Answer:**

Given a number 39 and need to find out the unit digit of the square of number.

Now, the square of number, n = n^{2}

Since, we have “9” as the unit place in the number “39”.

⇒ 9^{2} = 81

⇒ 1 will be the unit digit of square of 39.

Hence, the units’ digit of the square of 39 is 1.

**Question 2.**

What will be the units digit of the square of the following numbers?

297

**Answer:**

Given a number 297 and need to find out the unit digit of the square of number.

Now, the square of number, n = n^{2}

Since, we have “7” in the units place in the given number “297” we have to square the number 7

⇒ 7^{2} = 49 and 9 will be the units’ digit.

Hence, the units digit of the square of 297 is 9.

**Question 3.**

What will be the units digit of the square of the following numbers?

5125

**Answer:**

Given a number 5125 and need to find out the unit digit of the square of number.

Now, the square of number, n = n^{2}

Since, we have “5” in the units place in the given number “5125” we have to square the number 5

⇒ 5^{2} = 25 and 5 will be the units’ digit.

Hence, the units digit of the square of 5125 is 5.

**Question 4.**

What will be the units digit of the square of the following numbers?

7286

**Answer:**

Given a number 7286 and need to find out the unit digit of the square of number.

Now, the square of number, n = n^{2}

Since, we have “6” in the units place in the given number “7286” we have to square the number 6

⇒ 6^{2} = 36

⇒ 6 will be the units’ digit.

Hence, the units digit of the square of 7286 is 6.

**Question 5.**

What will be the units digit of the square of the following numbers?

8742

**Answer:**

Given a number 8742 and need to find out the unit digit of the square of number.

Now, the square of number, n = n^{2}

Since, we have “2” in the units place in the given number “8742” we have to square the number 2

⇒ 2^{2} = 4

⇒ 4will be the units’ digit.

Hence, the units digit of the square of 8742 is 4.

**Question 6.**

Which of the following numbers are perfect squares?

121

**Answer:**

Given, a number 121 and need to find out whether it is perfect square or not.

Now, we have the perfect square = , [T1] where m and n are integers

⇒ 121 is a perfect square as it can be expressed as 11 × 11 form the product of two equal integer.

⇒121 = 11 × [T2] 11

Hence, 121 is a perfect square.

**Question 7.**

Which of the following numbers are perfect squares?

136

**Answer:**

Given a number 136 and need to find out whether it is perfect square or not.

Now, we have the perfect square = , where m and n are integers

⇒ 136 is not a perfect square as it cannot be expressed as n × n form the product of two equal integer.

Hence, 136 is not a perfect square.

**Question 8.**

Which of the following numbers are perfect squares?

256

**Answer:**

Given a number 256 and need to find out whether it is perfect square or not.

Now, we have the perfect square = , where m and n are integers

⇒ 256 is a perfect square as it can be

expressed as 16 × 16 form the product of two equal integer.

⇒ 256 = 16 × 16

Hence, 256 is a perfect square.

**Question 9.**

Which of the following numbers are perfect squares?

321

**Answer:**

Given a number 321 and need to find out whether it is perfect square or not.

Now, we have the perfect square = , where m and n are integers

⇒ 321 is not a perfect square as it cannot be

expressed as n × n form the product of two equal integer.

Hence, 321 is not a perfect square.

**Question 10.**

Which of the following numbers are perfect squares?

600

**Answer:**

Given a number 600 and need to find out whether it is perfect square or not.

Now, we have the perfect square = , where m and n are integers

⇒ 600 is not a perfect square as it cannot be expressed as n × n form the product of two equal integer.

Hence, 600 is not a perfect square.

**Question 11.**

The following numbers are not perfect squares. Give reasons?

257

**Answer:**

Given a number 257 is not perfect square. Need to find out the reason.

Now, we have the perfect square = , where m and n are integers

⇒257 is not a perfect square as it cannot be expressed as n × n form the product of two equal

integer and the number that have 2,3,7 or 8 in the units place are not perfect squares.

Hence, the given number 257 is not a perfect square.

**Question 12.**

The following numbers are not perfect squares. Give reasons?

4592

**Answer:**

Given a number 4592 is not perfect square. Need to find out the reason.

Now, we have the perfect square = , where m and n are integers

⇒ 4592 is not a perfect square as it cannot be expressed as n × n form the product of two equal integer and the number that have 2,3,7 or 8 in the units place are not perfect squares.

Hence, the given number 4592 is not a perfect square.

**Question 13.**

The following numbers are not perfect squares. Give reasons?

2433

**Answer:**

Given a number 2433 is not perfect square. Need to find out the reason.

Now, we have the perfect square = , where m and n are integers

⇒ 2433 is not a perfect square as it cannot be expressed as n × n form the product of two equal integer and the number that have 2,3,7 or 8 in the units place are not perfect squares.

Hence, the given number 2433 is not a perfect square.

**Question 14.**

The following numbers are not perfect squares. Give reasons?

5050

**Answer:**

Given a number 5050 is not perfect square. Need to find out the reason.

Now, we have the perfect square = , where m and n are integers

⇒ 5050 is not a perfect square as it cannot be expressed as n × n form the product of two equal integer.

Hence, the given number 5050 is not a perfect square.

**Question 15.**

The following numbers are not perfect squares. Give reasons?

6098

**Answer:**

Given a number 6098 is not perfect square. Need to find out the reason.

Now, we have the perfect square = , where m and n are integers

⇒ 6098 is not a perfect square as it cannot be expressed as n × n form the product of two equal integer and the number that have 2,3,7 or 8 in the units place are not perfect squares.

Hence, the given number 6098 is not a perfect square.

**Question 16.**

Find whether the square of the following numbers are even or odd?

431

**Answer:**

Given a number 431 and need to find out whether it is even or odd

Now, the square of number, n = n^{2}

Consider units digit of a number 431 and square the units digit number.

⇒ Here, Units digit number is 1

⇒ Square of 1 = 1

⇒ Square the unit digit 1 = 1

⇒ 1 is a odd number

⇒ ∴ square of 431 will be again odd number

Hence, 431 is odd number.

**Question 17.**

Find whether the square of the following numbers are even or odd?

2826

**Answer:**

Given a number 2826 and need to find out whether it is even or odd

Now, the square of number, n = n^{2}

Consider units digit of a number 2826 and square the units digit number.

⇒ Here, Units digit number is 6

⇒ Square of 6 = 36

⇒ 6 is a even number

⇒ ∴ square of 2826 will be again even number

Hence, 2826 is even number.

**Question 18.**

Find whether the square of the following numbers are even or odd?

8204

**Answer:**

__Given:__ A number 8204 and need to find out whether it is even or odd

Now, the square of number, n = n^{2}

Consider units digit of a number 8204 and square the units digit number.

⇒ Here, Units digit number is 4

⇒ Square of 4 = 16

⇒ 4 is a even number

⇒ ∴ square of 8204 will be again even number

Hence, 8204 is even number.

**Question 19.**

Find whether the square of the following numbers are even or odd?

17779

**Answer:**

Given a number 17779 and need to find out whether it is even or odd

Now, the square of number, n = n^{2}

Consider units digit of a number 17779 and square

the units digit number.

⇒ Here, Units digit number is 9

⇒ Square of 9 = 81

⇒ 9 is a odd number

⇒ ∴ square of 17779 will be again odd number

Hence, 17779 is odd number.

**Question 20.**

Find whether the square of the following numbers are even or odd?

99998

**Answer:**

Given a number 99998 and need to find out whether it is even or odd

Now, the square of number, n = n^{2}

Consider units digit of a number 99998 and square the units digit number.

⇒ Here, Units digit number is 8

⇒ Square of 8 = 64

⇒ 8 is a even number

⇒ ∴ square of 99998 will be again even number

Hence, 99998 is even number.

**Question 21.**

How many numbers lie between the square of the following numbers

25; 26

**Answer:**

Given, two numbers 25 and 26 we need to find out the numbers lie between the squares of the given numbers.

Now, we have numbers lie between the square of n and (n + 1) as 2n i.e 2 × base of first number.

⇒ Numbers between squares of 25 and (25 + 1) = 2 × 25

⇒ 50

Hence, 50 numbers lies between the square of the given numbers.

**Question 22.**

How many numbers lie between the square of the following numbers

56; 57

**Answer:**

Given, two numbers 56 and 57 we need to find out the numbers lie between the squares of the given numbers.

Now, we have numbers lie between the square of n and (n + 1) as 2n i.e 2 × base of first number.

⇒ Numbers between squares of 56 and (56 + 1) = 2 × 56

⇒112

Hence, 112 numbers lies between the square of the given numbers.

**Question 23.**

How many numbers lie between the square of the following numbers

107;108

**Answer:**

Given, two numbers 107 and 108 we need to find out the numbers lie between the squares of the given numbers.

Now, we have numbers lie between the square of n and (n + 1) as 2n i.e 2 × base of first number.

⇒ Numbers between squares of 107 and (107 + 1) = 2 × 107

⇒ 214

Hence, 214 numbers lies between the square of the given numbers.

**Question 24.**

Without adding, find the sum of the following numbers

1 + 3 + 5 + 7 + 9 =

**Answer:**

Given, 5 consecutive odd numbers sum. To find out, the sum of consecutive odd numbers from “1” to “9” without adding.

We have sum of first n odd numbers = n^{2}.

Here, n = 5

⇒ 1 + 3 + 5 + 7 + 9 = 5^{2}

⇒ 1 + 3 + 5 + 7 + 9 = 25

Hence, the sum of 1 + 3 + 5 + 7 + 9 without adding = 25

**Question 25.**

Without adding, find the sum of the following numbers

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 =

**Answer:**

Given, 9 consecutive odd numbers sum. To find out, the sum of consecutive odd numbers from “1” to “17” without adding.

We have sum of first n odd numbers = n^{2}.

Here, n = 9

⇒ 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 9^{2}

⇒ 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81

Hence, the sum of 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 without adding = 81

**Question 26.**

Without adding, find the sum of the following numbers

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 =

**Answer:**

Given, 13 consecutive odd numbers sum. To find out, the sum of consecutive odd numbers from “1” to “25” without adding.

We have sum of first n odd numbers = n^{2}.

Here, n = 13

⇒ 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 = 13^{2}

⇒ 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 = 169

Hence, the sum of 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 without adding = 169

###### Exercise 6.2

**Question 1.**Find the square roots of the following numbers by Prime factorization method.

441

**Answer:**Given, a number as 441. We need to find out the square root using prime factorization method.

Step 1: Resolve the given number into prime factors, we get

⇒7 × 3 × 3 × 7

Step 2: Make pairs of equal factors, we get

⇒ (3 × 3) × (7 × 7)

Step 3: Choosing one factor out of every pair, we get

⇒3 × 7

⇒ 21

Hence, 21 is the square root of the given number 441 using prime factorization method

**Question 2.**Find the square roots of the following numbers by Prime factorization method.

784

**Answer:**Given, a number as 784. We need to find out the square root using prime factorization method.

Step 1: Resolve the given number into prime factors, we get

⇒2 × 2 × 2 × 2 × 7 × 7

Step 2: Make pairs of equal factors, we get

⇒ (2 × 2) × (2 × 2) × (7 × 7)

Step 3: Choosing one factor out of every pair, we get

⇒2 × 2 × 7

⇒ 28

Hence, 28 is the square root of the given number 784 using prime factorization method

**Question 3.**Find the square roots of the following numbers by Prime factorization method.

4096

**Answer:**Given, a number as 4096. We need to find out the square root using prime factorization method.

Step 1: Resolve the given number into prime factors, we get

⇒2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

Step 2: Make pairs of equal factors, we get

⇒ (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2)

Step 3: Choosing one factor out of every pair, we get

⇒2 × 2 × 2 × 2 × 2 × 2

⇒ 64

Hence, 64 is the square root of the given number 4096 using prime factorization method

**Question 4.**Find the square roots of the following numbers by Prime factorization method.

7056

**Answer:**Given, a number as 7056. We need to find out the square root using prime factorization method.

Step 1: Resolve the given number into prime factors, we get

⇒2 × 2 × 2 × 2 × 7 × 7 × 3 × 3

Step 2: Make pairs of equal factors, we get

⇒ (2 × 2) × (2 × 2) × (7 × 7) × (3 × 3)

Step 3: Choosing one factor out of every pair, we get

⇒2 × 2 × 7 × 3

⇒ 84

Hence, 84 is the square root of the given number 7056 using prime factorization method

**Question 5.**Find the smallest number by which 3645 must be multiplied to get a perfect square.

**Answer:**Given, a number as 3645. We need to find out a number which if multiplied by the given number we should get a perfect square.

Step 1: Resolve 3645 into prime factors

We get, 5 × 9 × 9 × 3 × 3

Step 2: Pair the factors obtained

⇒ (9 × 9) × (3 × 3) × 5

Step 3: multiply the number with the factor which is alone.

Here, 9,3 are in pair and 5 is alone.

So,we must multiply the given number by 5 to get a perfect square.

⇒ 3645 × 5 = 18225.

Hence, 5 should be multiplied to the given number “3645” to make it perfect square.

**Question 6.**Find the smallest number by which 2400 is to be multiplied to get a perfect square and also find the square root of the resulting number.

**Answer:**Given, a number as 2400. We need to find out a number which if multiplied by the given number we should get a perfect square.

To find out square root of the resulting number

Step 1: Resolve 2400 into prime factors

We get, 3 × 2 × 5 × 2 × 5 × 2 × 2 × 2

Step 2: Pair the factors obtained

⇒ (2 × 2) × (5 × 5) × (2 × 2) × 3 × 2

Step 3: multiply the number with the factor which is alone.

Here, 2,5 are in pairs and 3,2 are alone.

So, we must multiply the given number by 3 × 2 to get a perfect square.

⇒ 2400 × 3 × 2 = 2400 × 6 = 14400.

Step 4: The resulting number obtained is 14400

Square root is found out using common factors

⇒ = )

= 5 × 2 × 3 × 2 × 2

= 120

Hence, 6 should be multiplied to the given number “2400” to make it perfect square. The square root of the resulting number is 120.

**Question 7.**Find the smallest number by which 7776 is to be divided to get a perfect square.

**Answer:**Given, a number as 7776. We need to find out a number which if divided by the given number we should get a perfect square.

Step 1: Resolve 7776 into prime factors

We get, 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3

Step 2: Pair the factors obtained

⇒ (2 × 2) × (2 × 2) × (3 × 3) × (3 × 3) × 3 × 2

Step 3: divide the number with the factor which are alone.

Here, 2,3are in pairs and 3,2 are alone.

⇒ 3 × 2 = 6

⇒ = 1296.

Hence, 6 is the smallest number which is to be multiplied to the given number to get perfect square.

**Question 8.**1521 trees are planted in a garden in such a way that there are as many trees in each row as there are rows in the garden. Find the number of rows and number of trees in each row.

**Answer:**Given, 1521 trees are planted in a garden. We need to find out number of rows and number of trees.

Let us assume the number of trees in each row = x.

Since, we know that number of trees in each row = number of rows in the garden.

Total number of trees planted in a garden = x × x = x^{2}

Calculate x value.

∴ x^{2} = 1521

Calculate, prime factors for 1521.

We know from pairing the factors

⇒ x^{2} = (3 × 3) × (13 × 13)

⇒ x^{2} = 3 × 13

⇒X = 39

Hence, the number of trees in each row is 39 and number of rows in the garden is also 39.

**Question 9.**A school collected ` 2601 as fees from its students. If fee paid by each student and number students in the school were equal, how many students were there in the school?

**Answer:**Given, 2601 fees collected from the students. We need to find out number of students and fees paid by each student.

Let us assume the number of students in a school = x.

Since, we know that fee paid by each student = number of students in the school.

Total number of fees collected by the student = x × x = x^{2}

Calculate x value.

∴ x^{2} = 2601

Calculate prime factors for 2601.

We know from pairing the factors

⇒ x^{2} = (51 × 51)

⇒X = 51

Hence, the number of students in the school is 51.

**Question 10.**The product of two numbers is 1296. If one number is 16 times the other, find the two numbers?

**Answer:**Given, the product of two numbers = 1296 and one number is 16 times the other.

Let us, consider the number as x and other number as 16x.

Now, the product of two numbers = 1296

⇒ = 1296

⇒ = 1296

⇒ =

⇒ = 81

⇒ x =

⇒ x = 9

Another number is 16x = 144

∴ The numbers are 9 and 144

**Question 11.**7921 soldiers sat in an auditorium in such a way that there are as many soldiers in a row as there are rows in the auditorium. How many rows are there in the auditorium?

**Answer:**Given, 7921 soldiers in a auditorium. We need to find out number of rows and number of soldiers.

Let us assume the number of soldiers in each row = x.

Since, we know that number of soldiers in each row = number of rows in the auditorium.

Total number of soldiers = x × x = x^{2}

Calculate x value.

∴ x^{2} = 7921

Calculate prime factors for 7921.

We know from pairing the factors

⇒ x^{2} = (89 × 89)

⇒X = 89

Hence, number of rows in the auditorium is 89.

**Question 12.**The area of a square field is 5184 m^{2}. Find the area of a rectangular field, whose perimeter is equal to the perimeter of the square field and whose length is twice of its breadth.

**Answer:**Given, the area of a square field as 5184m^{2}

We need to find out the area of a rectangular field.

We know that perimeter of a rectangular field = perimeter of the square field and length = twice breadth.

Now, Area of a square field = 5184m^{2}

Area of a square = s × s, where s is side

⇒ = 72

Perimeter of a square field = 4 × s

= 4 × 72 = 288m

Perimeter of a square field = 288m = perimeter of a rectangular field.

Let breadth (b) = x and length (l) = 2x

We know that perimeter = 2 × (l + b)

= 2 × (2x + x)

= 2 × (3x)

= 6x

Here, 6x = 288m

⇒ 6x = 288

⇒ x =

⇒ x = 48m

Hence, breadth is 48m and length is 2 × 48 = 96m

**Question 1.**

Find the square roots of the following numbers by Prime factorization method.

441

**Answer:**

Given, a number as 441. We need to find out the square root using prime factorization method.

Step 1: Resolve the given number into prime factors, we get

⇒7 × 3 × 3 × 7

Step 2: Make pairs of equal factors, we get

⇒ (3 × 3) × (7 × 7)

Step 3: Choosing one factor out of every pair, we get

⇒3 × 7

⇒ 21

Hence, 21 is the square root of the given number 441 using prime factorization method

**Question 2.**

Find the square roots of the following numbers by Prime factorization method.

784

**Answer:**

Given, a number as 784. We need to find out the square root using prime factorization method.

Step 1: Resolve the given number into prime factors, we get

⇒2 × 2 × 2 × 2 × 7 × 7

Step 2: Make pairs of equal factors, we get

⇒ (2 × 2) × (2 × 2) × (7 × 7)

Step 3: Choosing one factor out of every pair, we get

⇒2 × 2 × 7

⇒ 28

Hence, 28 is the square root of the given number 784 using prime factorization method

**Question 3.**

Find the square roots of the following numbers by Prime factorization method.

4096

**Answer:**

Given, a number as 4096. We need to find out the square root using prime factorization method.

Step 1: Resolve the given number into prime factors, we get

⇒2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

Step 2: Make pairs of equal factors, we get

⇒ (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2)

Step 3: Choosing one factor out of every pair, we get

⇒2 × 2 × 2 × 2 × 2 × 2

⇒ 64

Hence, 64 is the square root of the given number 4096 using prime factorization method

**Question 4.**

Find the square roots of the following numbers by Prime factorization method.

7056

**Answer:**

Given, a number as 7056. We need to find out the square root using prime factorization method.

Step 1: Resolve the given number into prime factors, we get

⇒2 × 2 × 2 × 2 × 7 × 7 × 3 × 3

Step 2: Make pairs of equal factors, we get

⇒ (2 × 2) × (2 × 2) × (7 × 7) × (3 × 3)

Step 3: Choosing one factor out of every pair, we get

⇒2 × 2 × 7 × 3

⇒ 84

Hence, 84 is the square root of the given number 7056 using prime factorization method

**Question 5.**

Find the smallest number by which 3645 must be multiplied to get a perfect square.

**Answer:**

Given, a number as 3645. We need to find out a number which if multiplied by the given number we should get a perfect square.

Step 1: Resolve 3645 into prime factors

We get, 5 × 9 × 9 × 3 × 3

Step 2: Pair the factors obtained

⇒ (9 × 9) × (3 × 3) × 5

Step 3: multiply the number with the factor which is alone.

Here, 9,3 are in pair and 5 is alone.

So,we must multiply the given number by 5 to get a perfect square.

⇒ 3645 × 5 = 18225.

Hence, 5 should be multiplied to the given number “3645” to make it perfect square.

**Question 6.**

Find the smallest number by which 2400 is to be multiplied to get a perfect square and also find the square root of the resulting number.

**Answer:**

Given, a number as 2400. We need to find out a number which if multiplied by the given number we should get a perfect square.

To find out square root of the resulting number

Step 1: Resolve 2400 into prime factors

We get, 3 × 2 × 5 × 2 × 5 × 2 × 2 × 2

Step 2: Pair the factors obtained

⇒ (2 × 2) × (5 × 5) × (2 × 2) × 3 × 2

Step 3: multiply the number with the factor which is alone.

Here, 2,5 are in pairs and 3,2 are alone.

So, we must multiply the given number by 3 × 2 to get a perfect square.

⇒ 2400 × 3 × 2 = 2400 × 6 = 14400.

Step 4: The resulting number obtained is 14400

Square root is found out using common factors

⇒ = )

= 5 × 2 × 3 × 2 × 2

= 120

Hence, 6 should be multiplied to the given number “2400” to make it perfect square. The square root of the resulting number is 120.

**Question 7.**

Find the smallest number by which 7776 is to be divided to get a perfect square.

**Answer:**

Given, a number as 7776. We need to find out a number which if divided by the given number we should get a perfect square.

Step 1: Resolve 7776 into prime factors

We get, 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3

Step 2: Pair the factors obtained

⇒ (2 × 2) × (2 × 2) × (3 × 3) × (3 × 3) × 3 × 2

Step 3: divide the number with the factor which are alone.

Here, 2,3are in pairs and 3,2 are alone.

⇒ 3 × 2 = 6

⇒ = 1296.

Hence, 6 is the smallest number which is to be multiplied to the given number to get perfect square.

**Question 8.**

1521 trees are planted in a garden in such a way that there are as many trees in each row as there are rows in the garden. Find the number of rows and number of trees in each row.

**Answer:**

Given, 1521 trees are planted in a garden. We need to find out number of rows and number of trees.

Let us assume the number of trees in each row = x.

Since, we know that number of trees in each row = number of rows in the garden.

Total number of trees planted in a garden = x × x = x^{2}

Calculate x value.

∴ x^{2} = 1521

Calculate, prime factors for 1521.

We know from pairing the factors

⇒ x^{2} = (3 × 3) × (13 × 13)

⇒ x^{2} = 3 × 13

⇒X = 39

Hence, the number of trees in each row is 39 and number of rows in the garden is also 39.

**Question 9.**

A school collected ` 2601 as fees from its students. If fee paid by each student and number students in the school were equal, how many students were there in the school?

**Answer:**

Given, 2601 fees collected from the students. We need to find out number of students and fees paid by each student.

Let us assume the number of students in a school = x.

Since, we know that fee paid by each student = number of students in the school.

Total number of fees collected by the student = x × x = x^{2}

Calculate x value.

∴ x^{2} = 2601

Calculate prime factors for 2601.

We know from pairing the factors

⇒ x^{2} = (51 × 51)

⇒X = 51

Hence, the number of students in the school is 51.

**Question 10.**

The product of two numbers is 1296. If one number is 16 times the other, find the two numbers?

**Answer:**

Given, the product of two numbers = 1296 and one number is 16 times the other.

Let us, consider the number as x and other number as 16x.

Now, the product of two numbers = 1296

⇒ = 1296

⇒ = 1296

⇒ =

⇒ = 81

⇒ x =

⇒ x = 9

Another number is 16x = 144

∴ The numbers are 9 and 144

**Question 11.**

7921 soldiers sat in an auditorium in such a way that there are as many soldiers in a row as there are rows in the auditorium. How many rows are there in the auditorium?

**Answer:**

Given, 7921 soldiers in a auditorium. We need to find out number of rows and number of soldiers.

Let us assume the number of soldiers in each row = x.

Since, we know that number of soldiers in each row = number of rows in the auditorium.

Total number of soldiers = x × x = x^{2}

Calculate x value.

∴ x^{2} = 7921

Calculate prime factors for 7921.

We know from pairing the factors

⇒ x^{2} = (89 × 89)

⇒X = 89

Hence, number of rows in the auditorium is 89.

**Question 12.**

The area of a square field is 5184 m^{2}. Find the area of a rectangular field, whose perimeter is equal to the perimeter of the square field and whose length is twice of its breadth.

**Answer:**

Given, the area of a square field as 5184m^{2}

We need to find out the area of a rectangular field.

We know that perimeter of a rectangular field = perimeter of the square field and length = twice breadth.

Now, Area of a square field = 5184m^{2}

Area of a square = s × s, where s is side

⇒ = 72

Perimeter of a square field = 4 × s

= 4 × 72 = 288m

Perimeter of a square field = 288m = perimeter of a rectangular field.

Let breadth (b) = x and length (l) = 2x

We know that perimeter = 2 × (l + b)

= 2 × (2x + x)

= 2 × (3x)

= 6x

Here, 6x = 288m

⇒ 6x = 288

⇒ x =

⇒ x = 48m

Hence, breadth is 48m and length is 2 × 48 = 96m

###### Exercise 6.3

**Question 1.**Find the square roots of the following numbers by division method.

1089

**Answer:**Given, a number as 1089.

We need to find out the square root using division method

Step 1: Pair the given digit number, starting from units place to left as 10 and 89

Step 2: Find the largest number whose square is less than or equal to the first pair or single digit from left. Here we get 3 × 3 = 9

Step 3: Subtract the resulted value 9 from the number

Step 4 : Bring down the second pair 89 to the right of the remainder

Step 5: double the quotient 3

Step 6: Guess the possible value such that the product is equal to or less than new dividend

Hence, the square root of 1089 is 33

**Question 2.**Find the square roots of the following numbers by division method.

2304

**Answer:**Given, a number as 2304.

We need to find out the square root using division method

Step 1: Pair the given digit number, starting from units place to left as 23 and 04

Step 2: Find the largest number whose square is less than or equal to the first pair or single digit from left.

Step 3: Subtract the resulted value 16 from the given number

Step 4 : Bring down the second pair 04 to the right of the remainder

Step 5: double the quotient 4 + 4 = 8

Step 6: Guess the possible value such that the product is equal to or less than new dividend

Hence, the square root of 2304 is 48

**Question 3.**Find the square roots of the following numbers by division method.

7744

**Answer:**Given, a number as 7744.

We need to find out the square root using division method

Step 1: Pair the given digit number, starting from units place to left as 77 and 44

Step 2: Find the largest number whose square is less than or equal to the first pair or single digit from left.

Step 3: Subtract the resulted value 64 from the given number

Step 4 : Bring down the second pair 04 to the right of the remainder

Step 5: double the quotient 8 + 8 = 16

Step 6: Guess the possible value such that the product is equal to or less than new dividend

Hence, the square root of 7744 is 88

**Question 4.**Find the square roots of the following numbers by division method.

6084

**Answer:**Given, a number as 6084.

We need to find out the square root using division method

Step 1: Pair the given digit number, starting from units place to left as 60 and 84

Step 2: Find the largest number whose square is less than or equal to the first pair or single digit from left.

Step 3: Subtract the resulted value 49 from the given number

Step 4 : Bring down the second pair 84 to the right of the remainder

Step 5: double the quotient 7 + 7 = 14

Step 6: Guess the possible value such that the product is equal to or less than new dividend

Hence, the square root of 6084 is 78

**Question 5.**Find the square roots of the following numbers by division method.

9025

**Answer:**Given, a number as 9025.

We need to find out the square root using division method

Step 1: Pair the given digit number, starting from units place to left as 90 and 25

Step 2: Find the largest number whose square is less than or equal to the first pair or single digit from left.

Step 3: Subtract the resulted value 81 from the given number

Step 4 : Bring down the second pair 25 to the right of the remainder

Step 5: double the quotient 9 + 9 = 18

Step 6: Guess the possible value such that the product is equal to or less than new dividend

Hence, the square root of 9025 is 95

**Question 6.**Find the square roots of the following decimal numbers.

2.56

**Answer:**Given, a decimal number as 2.56.

We need to find out the square root using division method

Step 1: Pair the given digit number, starting from units place to left as 2 and 56

Step 2: Find the largest number whose square is less than or equal to the first pair or single digit from left.

Step 3: Subtract the resulted value 1 from the given number

Step 4 : Bring down the second pair .56 to the right of the remainder

Step 5: double the quotient 1 + 1 = 2

Step 6: Guess the possible value such that the product is equal to or less than new dividend

Hence, the square root of 2.56 is 1.6

**Question 7.**Find the square roots of the following decimal numbers.

18.49

**Answer:**Given, a decimal number as 18.49.

We need to find out the square root using division method

Step 1: Pair the given digit number, starting from units place to left as 18 and 49

Step 2: Find the largest number whose square is less than or equal to the first pair or single digit from left.

Step 3: Subtract the resulted value 16 from the given number

Step 4 : Bring down the second pair .49 to the right of the remainder

Step 5: double the quotient 4 + 4 = 8

Step 6: Guess the possible value such that the product is equal to or less than new dividend

Hence, the square root of 18.49 is 4.3

**Question 8.**Find the square roots of the following decimal numbers.

68.89

**Answer:**Given, a decimal number as 68.89.

We need to find out the square root using division method

Step 1: Pair the given digit number, starting from units place to left as 68 and 89

Step 2: Find the largest number whose square is less than or equal to the first pair or single digit from left.

Step 3: Subtract the resulted value 64 from the given number

Step 4 : Bring down the second pair .89 to the right of the remainder

Step 5: double the quotient 8 + 8 = 16

Step 6: Guess the possible value such that the product is equal to or less than new dividend

Hence, the square root of 68.89 is 8.3

**Question 9.**Find the square roots of the following decimal numbers.

84.64

**Answer:**Given, a decimal number as 84.64.

We need to find out the square root using division method

Step 1: Pair the given digit number, starting from units place to left as 84 and 64

Step 2: Find the largest number whose square is less than or equal to the first pair or single digit from left.

Step 3: Subtract the resulted value 81 from the given number

Step 4 : Bring down the second pair .64 to the right of the remainder

Step 5: double the quotient 9 + 9 = 18

Step 6: Guess the possible value such that the product is equal to or less than new dividend

Hence, the square root of 84.64 is 9.2

**Question 10.**Find the least number that is to be subtracted from 4000 to make it perfect square

**Answer:**Given, a number as 4000.

Need to find out the least number which must be subtracted to make it a perfect square.

Now, using by division method we get

That means if we subtract 31 from 4000 we get perfect square

⇒ 4000-31 = 3969.

= 63.

Hence, 31 must be subtracted from the given number to get a perfect square

**Question 11.**Find the length of the side of a square whose area is 4489 sq.cm.

**Answer:**Given, area of square as 4489 sq.cm

We need to find out the length.

Now, we know that area = l × l

Area = l^{2}

⇒ l^{2} = 4489

⇒ l =

=

⇒ l = 67 cm

Hence, the length of the side of a given square is 67cm

**Question 12.**A gardener wishes to plant 8289 plants in the form of a square and found that there were 8 plants left. How many plants were planted in each row?

**Answer:**Given, 8289 plants are to be planted in the form of square with 8 left to be planted.

We need to find out number of plants planted in each row.

Since, 8 plants are left from the given number of plants

= 8289 – 8

= 8281

To form a square number of plants planted in each row = number of rows.

Consider it as x

⇒ x × x = 8281

⇒X^{2} = 8281

X = 91.

Hence, 91 plants are planted in each row.

**Question 13.**Find the least perfect square with four digits.

**Answer:**We need to find out the least perfect square with four digits

Let us consider a four digit number as 1000 which is not a perfect square.

Now, we must find out a number when it is added to this number the resultant number will be a perfect square.

Using division method we get

We know that 1000 lies between 31^{2} and 32^{2}

⇒ 31^{2} < 1000 < 32^{2}

⇒ 32^{2} – 1000

= 1024 – 1000

= 24

24 must be added to 1000 we get 1024

Hence, the least perfect square with four digit is 1024

**Question 14.**Find the least number which must be added to 6412 to make it a perfect square?

**Answer:**Given, a number as 6412.

Need to find out the least number which must be added to make it a perfect square.

Now, using by division method we get

This shows that 6412 lies between 80^{2} and 81^{2}

⇒ 80^{2} < 6412 < 81^{2}

∴ the number to be added is 81^{2} – 6412

= 6561 – 6412

= 149

Hence, 149 must be added to make the given number as perfect square.

**Question 15.**Estimate the value of the following numbers to the nearest whole number

**Answer:**Given, a number as 97. We need to estimate the value of the number to the nearest whole number.

97 lies between 81 and 100

⇒ 9^{2} = 81 and 10^{2} = 100

∴ 81 < 97 <100

⇒ 9 < < 10

Thus, the approximate value of is 9

**Question 16.**Estimate the value of the following numbers to the nearest whole number

**Answer:**Given, a number as 250. We need to estimate the value of the number to the nearest whole number.

250 lies between 225 and 256

⇒ 15^{2} = 225 and 16^{2} = 256

∴ 225 < 250 <256

⇒ 15 < < 16

Thus, the approximate value of is 15

**Question 17.**Estimate the value of the following numbers to the nearest whole number

**Answer:**Given, a number as 780. We need to estimate the value of the number to the nearest whole number.

780 lies between 729 and 784

⇒ 27^{2} = 729 and 28^{2} = 784

∴ 729 < 780 <784

⇒ 27 < < 28

Thus, the approximate value of is 27

**Question 1.**

Find the square roots of the following numbers by division method.

1089

**Answer:**

Given, a number as 1089.

We need to find out the square root using division method

Step 1: Pair the given digit number, starting from units place to left as 10 and 89

Step 2: Find the largest number whose square is less than or equal to the first pair or single digit from left. Here we get 3 × 3 = 9

Step 3: Subtract the resulted value 9 from the number

Step 4 : Bring down the second pair 89 to the right of the remainder

Step 5: double the quotient 3

Step 6: Guess the possible value such that the product is equal to or less than new dividend

Hence, the square root of 1089 is 33

**Question 2.**

Find the square roots of the following numbers by division method.

2304

**Answer:**

Given, a number as 2304.

We need to find out the square root using division method

Step 1: Pair the given digit number, starting from units place to left as 23 and 04

Step 2: Find the largest number whose square is less than or equal to the first pair or single digit from left.

Step 3: Subtract the resulted value 16 from the given number

Step 4 : Bring down the second pair 04 to the right of the remainder

Step 5: double the quotient 4 + 4 = 8

Step 6: Guess the possible value such that the product is equal to or less than new dividend

Hence, the square root of 2304 is 48

**Question 3.**

Find the square roots of the following numbers by division method.

7744

**Answer:**

Given, a number as 7744.

We need to find out the square root using division method

Step 1: Pair the given digit number, starting from units place to left as 77 and 44

Step 2: Find the largest number whose square is less than or equal to the first pair or single digit from left.

Step 3: Subtract the resulted value 64 from the given number

Step 4 : Bring down the second pair 04 to the right of the remainder

Step 5: double the quotient 8 + 8 = 16

Step 6: Guess the possible value such that the product is equal to or less than new dividend

Hence, the square root of 7744 is 88

**Question 4.**

Find the square roots of the following numbers by division method.

6084

**Answer:**

Given, a number as 6084.

We need to find out the square root using division method

Step 1: Pair the given digit number, starting from units place to left as 60 and 84

Step 2: Find the largest number whose square is less than or equal to the first pair or single digit from left.

Step 3: Subtract the resulted value 49 from the given number

Step 4 : Bring down the second pair 84 to the right of the remainder

Step 5: double the quotient 7 + 7 = 14

Step 6: Guess the possible value such that the product is equal to or less than new dividend

Hence, the square root of 6084 is 78

**Question 5.**

Find the square roots of the following numbers by division method.

9025

**Answer:**

Given, a number as 9025.

We need to find out the square root using division method

Step 1: Pair the given digit number, starting from units place to left as 90 and 25

Step 3: Subtract the resulted value 81 from the given number

Step 4 : Bring down the second pair 25 to the right of the remainder

Step 5: double the quotient 9 + 9 = 18

Step 6: Guess the possible value such that the product is equal to or less than new dividend

Hence, the square root of 9025 is 95

**Question 6.**

Find the square roots of the following decimal numbers.

2.56

**Answer:**

Given, a decimal number as 2.56.

We need to find out the square root using division method

Step 1: Pair the given digit number, starting from units place to left as 2 and 56

Step 3: Subtract the resulted value 1 from the given number

Step 4 : Bring down the second pair .56 to the right of the remainder

Step 5: double the quotient 1 + 1 = 2

Step 6: Guess the possible value such that the product is equal to or less than new dividend

Hence, the square root of 2.56 is 1.6

**Question 7.**

Find the square roots of the following decimal numbers.

18.49

**Answer:**

Given, a decimal number as 18.49.

We need to find out the square root using division method

Step 1: Pair the given digit number, starting from units place to left as 18 and 49

Step 3: Subtract the resulted value 16 from the given number

Step 4 : Bring down the second pair .49 to the right of the remainder

Step 5: double the quotient 4 + 4 = 8

Step 6: Guess the possible value such that the product is equal to or less than new dividend

Hence, the square root of 18.49 is 4.3

**Question 8.**

Find the square roots of the following decimal numbers.

68.89

**Answer:**

Given, a decimal number as 68.89.

We need to find out the square root using division method

Step 1: Pair the given digit number, starting from units place to left as 68 and 89

Step 3: Subtract the resulted value 64 from the given number

Step 4 : Bring down the second pair .89 to the right of the remainder

Step 5: double the quotient 8 + 8 = 16

Step 6: Guess the possible value such that the product is equal to or less than new dividend

Hence, the square root of 68.89 is 8.3

**Question 9.**

Find the square roots of the following decimal numbers.

84.64

**Answer:**

Given, a decimal number as 84.64.

We need to find out the square root using division method

Step 1: Pair the given digit number, starting from units place to left as 84 and 64

Step 3: Subtract the resulted value 81 from the given number

Step 4 : Bring down the second pair .64 to the right of the remainder

Step 5: double the quotient 9 + 9 = 18

Step 6: Guess the possible value such that the product is equal to or less than new dividend

Hence, the square root of 84.64 is 9.2

**Question 10.**

Find the least number that is to be subtracted from 4000 to make it perfect square

**Answer:**

Given, a number as 4000.

Need to find out the least number which must be subtracted to make it a perfect square.

Now, using by division method we get

That means if we subtract 31 from 4000 we get perfect square

⇒ 4000-31 = 3969.

= 63.

Hence, 31 must be subtracted from the given number to get a perfect square

**Question 11.**

Find the length of the side of a square whose area is 4489 sq.cm.

**Answer:**

Given, area of square as 4489 sq.cm

We need to find out the length.

Now, we know that area = l × l

Area = l^{2}

⇒ l^{2} = 4489

⇒ l =

=

⇒ l = 67 cm

Hence, the length of the side of a given square is 67cm

**Question 12.**

A gardener wishes to plant 8289 plants in the form of a square and found that there were 8 plants left. How many plants were planted in each row?

**Answer:**

Given, 8289 plants are to be planted in the form of square with 8 left to be planted.

We need to find out number of plants planted in each row.

Since, 8 plants are left from the given number of plants

= 8289 – 8

= 8281

To form a square number of plants planted in each row = number of rows.

Consider it as x

⇒ x × x = 8281

⇒X^{2} = 8281

X = 91.

Hence, 91 plants are planted in each row.

**Question 13.**

Find the least perfect square with four digits.

**Answer:**

We need to find out the least perfect square with four digits

Let us consider a four digit number as 1000 which is not a perfect square.

Now, we must find out a number when it is added to this number the resultant number will be a perfect square.

Using division method we get

We know that 1000 lies between 31^{2} and 32^{2}

⇒ 31^{2} < 1000 < 32^{2}

⇒ 32^{2} – 1000

= 1024 – 1000

= 24

24 must be added to 1000 we get 1024

Hence, the least perfect square with four digit is 1024

**Question 14.**

Find the least number which must be added to 6412 to make it a perfect square?

**Answer:**

Given, a number as 6412.

Need to find out the least number which must be added to make it a perfect square.

Now, using by division method we get

This shows that 6412 lies between 80^{2} and 81^{2}

⇒ 80^{2} < 6412 < 81^{2}

∴ the number to be added is 81^{2} – 6412

= 6561 – 6412

= 149

Hence, 149 must be added to make the given number as perfect square.

**Question 15.**

Estimate the value of the following numbers to the nearest whole number

**Answer:**

Given, a number as 97. We need to estimate the value of the number to the nearest whole number.

97 lies between 81 and 100

⇒ 9^{2} = 81 and 10^{2} = 100

∴ 81 < 97 <100

⇒ 9 < < 10

Thus, the approximate value of is 9

**Question 16.**

Estimate the value of the following numbers to the nearest whole number

**Answer:**

Given, a number as 250. We need to estimate the value of the number to the nearest whole number.

250 lies between 225 and 256

⇒ 15^{2} = 225 and 16^{2} = 256

∴ 225 < 250 <256

⇒ 15 < < 16

Thus, the approximate value of is 15

**Question 17.**

Estimate the value of the following numbers to the nearest whole number

**Answer:**

Given, a number as 780. We need to estimate the value of the number to the nearest whole number.

780 lies between 729 and 784

⇒ 27^{2} = 729 and 28^{2} = 784

∴ 729 < 780 <784

⇒ 27 < < 28

Thus, the approximate value of is 27

###### Exercise 6.4

**Question 1.**Find the cubes of the following numbers

8

**Answer:**Given, a number as 8. We need to find out the cube of

the number.

Now, we know cube of a number as n^{3}

⇒ 8^{3}

⇒512

Hence, the cube of a given number 8 is 512.

**Question 2.**Find the cubes of the following numbers

16

**Answer:**Given, a number as 16. We need to find out the cube of

the number.

Now, we know cube of a number as n^{3}

⇒ 16^{3}

⇒4096

Hence, the cube of a given number 16 is 4096.

**Question 3.**Find the cubes of the following numbers

21

**Answer:**Given, a number as 21. We need to find out the cube of

the number.

Now, we know cube of a number as n^{3}

⇒ 21^{3}

⇒9261

Hence, the cube of a given number 21 is 9261.

**Question 4.**Find the cubes of the following numbers

30

**Answer:**Given, a number as 30. We need to find out the cube of

the number.

Now, we know cube of a number as n^{3}

⇒ 30^{3}

⇒27000

Hence, the cube of a given number 30 is 27000.

**Question 5.**Test whether the given numbers are perfect cubes or not.

243

**Answer:**Given, a number 243 and need to find out whether it is perfect cube or not.

Now, we have the perfect cube = , where m and n are integers

⇒ 243 is not a perfect cube as it cannot be expressed as n × n × n form the product of three equal integer.

Hence, 243 is not a perfect cube.

**Question 6.**Test whether the given numbers are perfect cubes or not.

516

**Answer:**Given, a number 516 and need to find out whether it is perfect cube or not.

Now, we have the perfect cube = , where m and n are integers

⇒ 516 is not a perfect cube as it cannot be expressed as n × n × n form the product of three equal integer.

Hence, 516 is not a perfect cube

**Question 7.**Test whether the given numbers are perfect cubes or not.

729

**Answer:**Given, a number 729 and need to find out whether it is perfect cube or not.

Now, we have the perfect cube = , where m and n are integers

⇒ 729 is a perfect cube as it can be expressed as

n × n × n form the product of three equal

integer.

⇒ 729 = 9 × 9 × 9

Hence, 729 is a perfect cube.

**Question 8.**Test whether the given numbers are perfect cubes or not.

8000

**Answer:**Given, a number 8000 and need to find out whether it is perfect cube or not.

Now, we have the perfect cube = , where m and n are integers

⇒ 8000 is a perfect cube as it can be expressed as

n × n × n form the product of three equal

integer.

⇒ 8000 = 20 × 20 × 20

Hence, 8000 is a perfect cube.

**Question 9.**Test whether the given numbers are perfect cubes or not.

2700

**Answer:**Given, a number 2700 and need to find out whether it is perfect cube or not.

Now, we have the perfect cube = , where m and n are integers

⇒ 2700 is not a perfect cube as it cannot be expressed as n × n × n form the product of three equal integer.

Hence, 2700 is not a perfect cube

**Question 10.**Find the smallest number by which 8788 must be multiplied to obtain a perfect cube?

**Answer:**Given, a number as 8788. We need to find out a number which if multiplied by the given number we get a perfect cube.

Step 1: Resolve 8788 into prime factors

We get, 2 × 2 × 13 × 13 × 13

Step 2: Pair the factors obtained in the group of three

⇒ (2 × 2) × (13 × 13 × 13)

Step 3: multiply the number with the factor which is alone.

Here, 13 is in group of three and 2 is alone.

⇒ 8788 × 2 = 17576

Hence, 2 is the smallest number which is to be multiplied to the given number for perfect cube.

**Question 11.**What smallest number should 7803 be multiplied with so that the product becomes a perfect cube?

**Answer:**Given, a number as 7803. We need to find out a number which if multiplied by the given number we get a perfect cube.

Step 1: Resolve 7803 into prime factors

We get, 3 × 3 × 3 × 17 × 17

Step 2: Pair the factors obtained in the group of three

⇒ (3 × 3 × 3) × (17 × 17)

Step 3: multiply the number with the factor which is alone.

Here, 3 is in group of three and 17 is alone.

⇒ 7803 × 17 = 132651

Hence, 17 is the smallest number which is to be multiplied to the given number for perfect cube.

**Question 12.**Find the smallest number by which 8640 must be divided so that the quotient is a perfect cube?

**Answer:**Given, a number as 8640. We need to find out a number which if divided by the given number we get quotient as a perfect cube.

Step 1: Resolve 8640 into prime factors

We get, 2 × 2 × 2 × 2 × 2 × 2 × 5 × 3 × 3 × 3

Step 2: Pair the factors obtained in the group of three

⇒ (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3) × 5

Step 3: divide the number with the factor which is alone.

Here, 2, 3 is in group of three and 5 is alone.

⇒ = 1728

Hence, 5 is the smallest number which is to be divided to the given number for perfect cube.

**Question 13.**Ravi made a cuboid of plasticine of dimensions 12cm, 8cm and 3cm. How many minimum numbers of such cuboids will be needed to form a cube?

**Answer:**Given, a cubiod with sides as 12cm, 8 cm and 3cm

To find out number of cuboids required to form a cube.

Volume of the cubiod = 12 × 8 × 3 cm^{3}

The cube is formed by stacking many such cuboids and it will have side lengths which are multiple of 12,8,3 cm

The smallest such cube will have a side length of LCM(12,8,3)

LCM is found out as follows:

LCM is 3 × 4 × 2 = 24 cm

And a volume = (24 × 24 × 24)cm

∴ Number of cuboids to fit in a cube =

= 2 × 24

= 48

Hence, 48 cuboids are needed to form a cube

**Question 14.**Find the smallest prime number dividing the sum 3^{11} + 5^{13}.

**Answer:**Given, the sum of 3^{11} and 5^{13}

We need to find out smallest prime numbers.

We know that sum of two odd numbers is even and any even number is divisible by 2.

Hence, the sum of 3^{11} and 5^{13} is divisible by 2 which is the smallest prime number.

**Question 1.**

Find the cubes of the following numbers

8

**Answer:**

Given, a number as 8. We need to find out the cube of

the number.

Now, we know cube of a number as n^{3}

⇒ 8^{3}

⇒512

Hence, the cube of a given number 8 is 512.

**Question 2.**

Find the cubes of the following numbers

16

**Answer:**

Given, a number as 16. We need to find out the cube of

the number.

Now, we know cube of a number as n^{3}

⇒ 16^{3}

⇒4096

Hence, the cube of a given number 16 is 4096.

**Question 3.**

Find the cubes of the following numbers

21

**Answer:**

Given, a number as 21. We need to find out the cube of

the number.

Now, we know cube of a number as n^{3}

⇒ 21^{3}

⇒9261

Hence, the cube of a given number 21 is 9261.

**Question 4.**

Find the cubes of the following numbers

30

**Answer:**

Given, a number as 30. We need to find out the cube of

the number.

Now, we know cube of a number as n^{3}

⇒ 30^{3}

⇒27000

Hence, the cube of a given number 30 is 27000.

**Question 5.**

Test whether the given numbers are perfect cubes or not.

243

**Answer:**

Given, a number 243 and need to find out whether it is perfect cube or not.

Now, we have the perfect cube = , where m and n are integers

⇒ 243 is not a perfect cube as it cannot be expressed as n × n × n form the product of three equal integer.

Hence, 243 is not a perfect cube.

**Question 6.**

Test whether the given numbers are perfect cubes or not.

516

**Answer:**

Given, a number 516 and need to find out whether it is perfect cube or not.

Now, we have the perfect cube = , where m and n are integers

⇒ 516 is not a perfect cube as it cannot be expressed as n × n × n form the product of three equal integer.

Hence, 516 is not a perfect cube

**Question 7.**

Test whether the given numbers are perfect cubes or not.

729

**Answer:**

Given, a number 729 and need to find out whether it is perfect cube or not.

Now, we have the perfect cube = , where m and n are integers

⇒ 729 is a perfect cube as it can be expressed as

n × n × n form the product of three equal

integer.

⇒ 729 = 9 × 9 × 9

Hence, 729 is a perfect cube.

**Question 8.**

Test whether the given numbers are perfect cubes or not.

8000

**Answer:**

Given, a number 8000 and need to find out whether it is perfect cube or not.

Now, we have the perfect cube = , where m and n are integers

⇒ 8000 is a perfect cube as it can be expressed as

n × n × n form the product of three equal

integer.

⇒ 8000 = 20 × 20 × 20

Hence, 8000 is a perfect cube.

**Question 9.**

Test whether the given numbers are perfect cubes or not.

2700

**Answer:**

Given, a number 2700 and need to find out whether it is perfect cube or not.

Now, we have the perfect cube = , where m and n are integers

⇒ 2700 is not a perfect cube as it cannot be expressed as n × n × n form the product of three equal integer.

Hence, 2700 is not a perfect cube

**Question 10.**

Find the smallest number by which 8788 must be multiplied to obtain a perfect cube?

**Answer:**

Given, a number as 8788. We need to find out a number which if multiplied by the given number we get a perfect cube.

Step 1: Resolve 8788 into prime factors

We get, 2 × 2 × 13 × 13 × 13

Step 2: Pair the factors obtained in the group of three

⇒ (2 × 2) × (13 × 13 × 13)

Step 3: multiply the number with the factor which is alone.

Here, 13 is in group of three and 2 is alone.

⇒ 8788 × 2 = 17576

Hence, 2 is the smallest number which is to be multiplied to the given number for perfect cube.

**Question 11.**

What smallest number should 7803 be multiplied with so that the product becomes a perfect cube?

**Answer:**

Given, a number as 7803. We need to find out a number which if multiplied by the given number we get a perfect cube.

Step 1: Resolve 7803 into prime factors

We get, 3 × 3 × 3 × 17 × 17

Step 2: Pair the factors obtained in the group of three

⇒ (3 × 3 × 3) × (17 × 17)

Step 3: multiply the number with the factor which is alone.

Here, 3 is in group of three and 17 is alone.

⇒ 7803 × 17 = 132651

Hence, 17 is the smallest number which is to be multiplied to the given number for perfect cube.

**Question 12.**

Find the smallest number by which 8640 must be divided so that the quotient is a perfect cube?

**Answer:**

Given, a number as 8640. We need to find out a number which if divided by the given number we get quotient as a perfect cube.

Step 1: Resolve 8640 into prime factors

We get, 2 × 2 × 2 × 2 × 2 × 2 × 5 × 3 × 3 × 3

Step 2: Pair the factors obtained in the group of three

⇒ (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3) × 5

Step 3: divide the number with the factor which is alone.

Here, 2, 3 is in group of three and 5 is alone.

⇒ = 1728

Hence, 5 is the smallest number which is to be divided to the given number for perfect cube.

**Question 13.**

Ravi made a cuboid of plasticine of dimensions 12cm, 8cm and 3cm. How many minimum numbers of such cuboids will be needed to form a cube?

**Answer:**

Given, a cubiod with sides as 12cm, 8 cm and 3cm

To find out number of cuboids required to form a cube.

Volume of the cubiod = 12 × 8 × 3 cm^{3}

The cube is formed by stacking many such cuboids and it will have side lengths which are multiple of 12,8,3 cm

The smallest such cube will have a side length of LCM(12,8,3)

LCM is found out as follows:

LCM is 3 × 4 × 2 = 24 cm

And a volume = (24 × 24 × 24)cm

∴ Number of cuboids to fit in a cube =

= 2 × 24

= 48

Hence, 48 cuboids are needed to form a cube

**Question 14.**

Find the smallest prime number dividing the sum 3^{11} + 5^{13}.

**Answer:**

Given, the sum of 3^{11} and 5^{13}

We need to find out smallest prime numbers.

We know that sum of two odd numbers is even and any even number is divisible by 2.

Hence, the sum of 3^{11} and 5^{13} is divisible by 2 which is the smallest prime number.

###### Exercise 6.5

**Question 1.**Find the cube root of the following numbers by prime factorization method.

343

**Answer:**Given, a number as 343. We need to find out the cube root using prime factorization method.

Step 1: Resolve the given number into prime factors, we get

343 = 7 × 7 × 7 × 1

Step 2: Make pairs of equal factors, we get

⇒ (7 × 7 × 7) × 1

Step 3: Choosing one factor out of every pair, we get

⇒7

Hence, 7 is the cube root of the given number 343 using prime factorization method

**Question 2.**Find the cube root of the following numbers by prime factorization method.

729

**Answer:**Given, a number as 729. We need to find out the cube root using prime factorization method.

Step 1: Resolve the given number into prime factors, we get

⇒3 × 3 × 3 × 3 × 3 × 3

Step 2: Make pairs of equal factors, we get

⇒ (3 × 3 × 3) × (3 × 3 × 3)

Step 3: Choosing one factor out of every pair, we get

⇒3 × 3

⇒ 9

Hence, 9 is the cube root of the given number 729 using prime factorization method

**Question 3.**Find the cube root of the following numbers by prime factorization method.

1331

**Answer:**Given, a number as 1331. We need to find out the cube root using prime factorization method.

Step 1: Resolve the given number into prime factors, we get

⇒ 11 × 11 × 11

Step 2: Make pairs of equal factors, we get

⇒ (11 × 11 × 11)

Step 3: Choosing one factor out of every pair, we get

⇒ 11

Hence, 11 is the cube root of the given number 1331 using prime factorization method

**Question 4.**Find the cube root of the following numbers by prime factorization method.

2744

**Answer:**Given, a number as 2744. We need to find out the cube root using prime factorization method.

Step 1: Resolve the given number into prime factors, we get

⇒2 × 2 × 2 × 7 × 7 × 7

Step 2: Make pairs of equal factors, we get

⇒ (2 × 2 × 2) × (7 × 7 × 7)

Step 3: Choosing one factor out of every pair, we get

⇒ 2 × 7 = 14

Hence, 14 is the cube root of the given number 2744 using prime factorization method

**Question 5.**Find the cube root of the following numbers through estimation?

512

**Answer:**Given number as 512. We need to find out the cube root using estimation method.

Step 1: start making groups of three digits starting from the units place.

⇒ 512 as first group and it has no second group

Step 2: first group will give us the units digit of the cube root.

⇒ 512 ends with 2, cube of 2 = 2^{3} = 8

∴ 8 will go in units place

Step 3: Now, we do not have second group to calculate

8 becomes the required cube root.

∴ = 8

Hence, the cube root of 512 using estimation method is 8

**Question 6.**Find the cube root of the following numbers through estimation?

2197

**Answer:**Given number as 2197. We need to find out the cube root using estimation method.

Step 1: start making groups of three digits starting from the units place.

⇒ 197 as first group and 2 as second group

Step 2: first group will give us the units digit of the cube root.

⇒ 197 ends with 7, cube of 7 = 7^{3} = 343

∴ 3 will go in units place

Step 3: Now, take second group. i.e 2

⇒ we know 1^{3} < 2 < 3^{3}

As the smallest number is 1, it becomes the tens place of the required cube root.

∴ = 13

Hence, the cube root of 2197 using estimation method is 13

**Question 7.**Find the cube root of the following numbers through estimation?

3375

**Answer:**Given number as 3375. We need to find out the cube root using estimation method.

Step 1: start making groups of three digits starting from the units place.

⇒ 375 as first group and 3 as second group

Step 2: first group will give us the units digit of the cube root.

⇒ 375 ends with 5, cube of 5 = 5^{3} = 125

∴ 5 will go in units place

Step 3: Now, take second group. i.e 3

⇒ we know 1^{3} < 3 < 2^{3}

As the smallest number is 1, it becomes the tens place of the required cube root.

∴ = 15

Hence, the cube root of 3375 using estimation method is 15

**Question 8.**Find the cube root of the following numbers through estimation?

5832

**Answer:**Given number as 5832. We need to find out the cube root using estimation method.

Step 1: start making groups of three digits starting from the units place.

⇒ 832 as first group and 5 as second group

Step 2: first group will give us the units digit of the cube root.

⇒ 832 ends with 2, cube of 2 = 2^{3} = 8

∴ 8 will go in units place

Step 3: Now, take second group. i.e 5

⇒ we know 1^{3} < 5 < 2^{3}

As the smallest number is 1, it becomes the tens place of the required cube root.

∴ = 18

Hence, the cube root of 5832 using estimation method is 18

**Question 9.**State true or false?

Cube of an even number is an odd number

**Answer:**The given statement is false

Consider cube of even numbers

⇒ 2^{3} = 8, 4^{3} = 64,6^{3} = 216 all are even numbers.

Hence, it is false that the cube of an even number is an odd number

**Question 10.**State true or false?

A perfect cube may end with two zeros

**Answer:**The given statement is false

Since, a perfect cube ends with three zeros

⇒ Consider 10^{3} = 1000,20^{3} = 8000

Hence, it is false that a perfect cube may end with two zeros.

**Question 11.**State true or false?

If a number ends with 5, then its cube ends with 5

**Answer:**The given statement is true.

Consider a number 5

⇒ cube of 5 = 5^{3} = 125

Hence, it is true that number ends with 5, then its cube ends with 5

**Question 12.**State true or false?

Cube of a number ending with zero has three zeros at its right

**Answer:**The given statement is true.

⇒ Consider a number ending with zero as 10

⇒ cube of that number is 10^{3} = 1000 (Has three zeros at its right)

Hence, it is true that a number ending with zero has three zeros at its right.

**Question 13.**State true or false?

The cube of a single digit number may be a single digit number.

**Answer:**The given statement is true.

Consider a single digit number “2” which is second smallest single digit number.

Cube of 2 = 2^{3} = 8.

⇒ 8 is a single digit number

Hence, it is true that the cube of a single digit number may be a single digit number.

**Question 14.**State true or false?

There is no perfect cube which ends with 8

**Answer:**The given statement is false

Since, cube of 2 = 2^{3} = 8

Hence, it is false that no perfect cube ends with 8

**Question 15.**State true or false?

The cube of a two-digit number may be a three-digit number.

**Answer:**The given statement is false.

⇒ Let us consider, the smallest two-digit number 10

⇒ cube of 10 = 10^{3}

⇒ 1000 (not a three-digit number)

Hence, it is false that the cube of a two-digit number may be a three-digit number.

**Question 16.**Find the two-digit number which a square number is and also a cubic number.

**Answer:**Need to find out a two-digit number which is a square number and also a cubic number.

⇒ A number which is a square must equal to = x^{2}

⇒ A number which is a cube must equal to = y^{3}

⇒ Number must be sixth power of an integer = z^{6}

∴ we can have x = z^{3} and y = z^{2} so x^{2} = z^{6} and y^{3} = z^{6}

By trial and error method 1^{6} = 1 , 2^{6} = 64 and 3^{6} = 729 (need two digit number).

So, 64 is the number.

⇒ 8^{2} = 64 = 4^{3}

Hence, 64 is the two-digit number which is a square number and also cubic number.

**Question 1.**

Find the cube root of the following numbers by prime factorization method.

343

**Answer:**

Given, a number as 343. We need to find out the cube root using prime factorization method.

Step 1: Resolve the given number into prime factors, we get

343 = 7 × 7 × 7 × 1

Step 2: Make pairs of equal factors, we get

⇒ (7 × 7 × 7) × 1

Step 3: Choosing one factor out of every pair, we get

⇒7

Hence, 7 is the cube root of the given number 343 using prime factorization method

**Question 2.**

Find the cube root of the following numbers by prime factorization method.

729

**Answer:**

Given, a number as 729. We need to find out the cube root using prime factorization method.

Step 1: Resolve the given number into prime factors, we get

⇒3 × 3 × 3 × 3 × 3 × 3

Step 2: Make pairs of equal factors, we get

⇒ (3 × 3 × 3) × (3 × 3 × 3)

Step 3: Choosing one factor out of every pair, we get

⇒3 × 3

⇒ 9

Hence, 9 is the cube root of the given number 729 using prime factorization method

**Question 3.**

Find the cube root of the following numbers by prime factorization method.

1331

**Answer:**

Given, a number as 1331. We need to find out the cube root using prime factorization method.

Step 1: Resolve the given number into prime factors, we get

⇒ 11 × 11 × 11

Step 2: Make pairs of equal factors, we get

⇒ (11 × 11 × 11)

Step 3: Choosing one factor out of every pair, we get

⇒ 11

Hence, 11 is the cube root of the given number 1331 using prime factorization method

**Question 4.**

Find the cube root of the following numbers by prime factorization method.

2744

**Answer:**

Given, a number as 2744. We need to find out the cube root using prime factorization method.

Step 1: Resolve the given number into prime factors, we get

⇒2 × 2 × 2 × 7 × 7 × 7

Step 2: Make pairs of equal factors, we get

⇒ (2 × 2 × 2) × (7 × 7 × 7)

Step 3: Choosing one factor out of every pair, we get

⇒ 2 × 7 = 14

Hence, 14 is the cube root of the given number 2744 using prime factorization method

**Question 5.**

Find the cube root of the following numbers through estimation?

512

**Answer:**

Given number as 512. We need to find out the cube root using estimation method.

Step 1: start making groups of three digits starting from the units place.

⇒ 512 as first group and it has no second group

Step 2: first group will give us the units digit of the cube root.

⇒ 512 ends with 2, cube of 2 = 2^{3} = 8

∴ 8 will go in units place

Step 3: Now, we do not have second group to calculate

8 becomes the required cube root.

∴ = 8

Hence, the cube root of 512 using estimation method is 8

**Question 6.**

Find the cube root of the following numbers through estimation?

2197

**Answer:**

Given number as 2197. We need to find out the cube root using estimation method.

Step 1: start making groups of three digits starting from the units place.

⇒ 197 as first group and 2 as second group

Step 2: first group will give us the units digit of the cube root.

⇒ 197 ends with 7, cube of 7 = 7^{3} = 343

∴ 3 will go in units place

Step 3: Now, take second group. i.e 2

⇒ we know 1^{3} < 2 < 3^{3}

As the smallest number is 1, it becomes the tens place of the required cube root.

∴ = 13

Hence, the cube root of 2197 using estimation method is 13

**Question 7.**

Find the cube root of the following numbers through estimation?

3375

**Answer:**

Given number as 3375. We need to find out the cube root using estimation method.

Step 1: start making groups of three digits starting from the units place.

⇒ 375 as first group and 3 as second group

Step 2: first group will give us the units digit of the cube root.

⇒ 375 ends with 5, cube of 5 = 5^{3} = 125

∴ 5 will go in units place

Step 3: Now, take second group. i.e 3

⇒ we know 1^{3} < 3 < 2^{3}

As the smallest number is 1, it becomes the tens place of the required cube root.

∴ = 15

Hence, the cube root of 3375 using estimation method is 15

**Question 8.**

Find the cube root of the following numbers through estimation?

5832

**Answer:**

Given number as 5832. We need to find out the cube root using estimation method.

Step 1: start making groups of three digits starting from the units place.

⇒ 832 as first group and 5 as second group

Step 2: first group will give us the units digit of the cube root.

⇒ 832 ends with 2, cube of 2 = 2^{3} = 8

∴ 8 will go in units place

Step 3: Now, take second group. i.e 5

⇒ we know 1^{3} < 5 < 2^{3}

As the smallest number is 1, it becomes the tens place of the required cube root.

∴ = 18

Hence, the cube root of 5832 using estimation method is 18

**Question 9.**

State true or false?

Cube of an even number is an odd number

**Answer:**

The given statement is false

Consider cube of even numbers

⇒ 2^{3} = 8, 4^{3} = 64,6^{3} = 216 all are even numbers.

Hence, it is false that the cube of an even number is an odd number

**Question 10.**

State true or false?

A perfect cube may end with two zeros

**Answer:**

The given statement is false

Since, a perfect cube ends with three zeros

⇒ Consider 10^{3} = 1000,20^{3} = 8000

Hence, it is false that a perfect cube may end with two zeros.

**Question 11.**

State true or false?

If a number ends with 5, then its cube ends with 5

**Answer:**

The given statement is true.

Consider a number 5

⇒ cube of 5 = 5^{3} = 125

Hence, it is true that number ends with 5, then its cube ends with 5

**Question 12.**

State true or false?

Cube of a number ending with zero has three zeros at its right

**Answer:**

The given statement is true.

⇒ Consider a number ending with zero as 10

⇒ cube of that number is 10^{3} = 1000 (Has three zeros at its right)

Hence, it is true that a number ending with zero has three zeros at its right.

**Question 13.**

State true or false?

The cube of a single digit number may be a single digit number.

**Answer:**

The given statement is true.

Consider a single digit number “2” which is second smallest single digit number.

Cube of 2 = 2^{3} = 8.

⇒ 8 is a single digit number

Hence, it is true that the cube of a single digit number may be a single digit number.

**Question 14.**

State true or false?

There is no perfect cube which ends with 8

**Answer:**

The given statement is false

Since, cube of 2 = 2^{3} = 8

Hence, it is false that no perfect cube ends with 8

**Question 15.**

State true or false?

The cube of a two-digit number may be a three-digit number.

**Answer:**

The given statement is false.

⇒ Let us consider, the smallest two-digit number 10

⇒ cube of 10 = 10^{3}

⇒ 1000 (not a three-digit number)

Hence, it is false that the cube of a two-digit number may be a three-digit number.

**Question 16.**

Find the two-digit number which a square number is and also a cubic number.

**Answer:**

Need to find out a two-digit number which is a square number and also a cubic number.

⇒ A number which is a square must equal to = x^{2}

⇒ A number which is a cube must equal to = y^{3}

⇒ Number must be sixth power of an integer = z^{6}

∴ we can have x = z^{3} and y = z^{2} so x^{2} = z^{6} and y^{3} = z^{6}

By trial and error method 1^{6} = 1 , 2^{6} = 64 and 3^{6} = 729 (need two digit number).

So, 64 is the number.

⇒ 8^{2} = 64 = 4^{3}

Hence, 64 is the two-digit number which is a square number and also cubic number.