### Square Roots And Cube Roots Class 8th Mathematics AP Board Solution

##### Question 1.Find the cube root of the following numbers by prime factorization method.343Answer: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 × 1Step 2: Make pairs of equal factors, we get⇒ (7 × 7 × 7) × 1Step 3: Choosing one factor out of every pair, we get⇒7Hence, 7 is the cube root of the given number 343 using prime factorization methodQuestion 2.Find the cube root of the following numbers by prime factorization method.729Answer: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 × 3Step 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⇒ 9Hence, 9 is the cube root of the given number 729 using prime factorization methodQuestion 3.Find the cube root of the following numbers by prime factorization method.1331Answer: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 × 11Step 2: Make pairs of equal factors, we get⇒ (11 × 11 × 11)Step 3: Choosing one factor out of every pair, we get⇒ 11Hence, 11 is the cube root of the given number 1331 using prime factorization methodQuestion 4.Find the cube root of the following numbers by prime factorization method.2744Answer: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 × 7Step 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 = 14Hence, 14 is the cube root of the given number 2744 using prime factorization methodQuestion 5.Find the cube root of the following numbers through estimation?512Answer: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 groupStep 2: first group will give us the units digit of the cube root.⇒ 512 ends with 2, cube of 2 = 23 = 8∴ 8 will go in units placeStep 3: Now, we do not have second group to calculate8 becomes the required cube root.∴ = 8Hence, the cube root of 512 using estimation method is 8Question 6.Find the cube root of the following numbers through estimation?2197Answer: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 groupStep 2: first group will give us the units digit of the cube root.⇒ 197 ends with 7, cube of 7 = 73 = 343∴ 3 will go in units placeStep 3: Now, take second group. i.e 2⇒ we know 13 < 2 < 33As the smallest number is 1, it becomes the tens place of the required cube root.∴ = 13Hence, the cube root of 2197 using estimation method is 13Question 7.Find the cube root of the following numbers through estimation?3375Answer: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 groupStep 2: first group will give us the units digit of the cube root.⇒ 375 ends with 5, cube of 5 = 53 = 125∴ 5 will go in units placeStep 3: Now, take second group. i.e 3⇒ we know 13 < 3 < 23As the smallest number is 1, it becomes the tens place of the required cube root.∴ = 15Hence, the cube root of 3375 using estimation method is 15Question 8.Find the cube root of the following numbers through estimation?5832Answer: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 groupStep 2: first group will give us the units digit of the cube root.⇒ 832 ends with 2, cube of 2 = 23 = 8∴ 8 will go in units placeStep 3: Now, take second group. i.e 5⇒ we know 13 < 5 < 23As the smallest number is 1, it becomes the tens place of the required cube root.∴ = 18Hence, the cube root of 5832 using estimation method is 18Question 9.State true or false?Cube of an even number is an odd numberAnswer:The given statement is falseConsider cube of even numbers⇒ 23 = 8, 43 = 64,63 = 216 all are even numbers.Hence, it is false that the cube of an even number is an odd numberQuestion 10.State true or false?A perfect cube may end with two zerosAnswer:The given statement is falseSince, a perfect cube ends with three zeros⇒ Consider 103 = 1000,203 = 8000Hence, 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 5Answer:The given statement is true.Consider a number 5⇒ cube of 5 = 53 = 125Hence, it is true that number ends with 5, then its cube ends with 5Question 12.State true or false?Cube of a number ending with zero has three zeros at its rightAnswer:The given statement is true.⇒ Consider a number ending with zero as 10⇒ cube of that number is 103 = 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 = 23 = 8.⇒ 8 is a single digit numberHence, 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 8Answer:The given statement is falseSince, cube of 2 = 23 = 8Hence, it is false that no perfect cube ends with 8Question 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 = 103⇒ 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 = x2⇒ A number which is a cube must equal to = y3⇒ Number must be sixth power of an integer = z6∴ we can have x = z3 and y = z2 so x2 = z6 and y3 = z6By trial and error method 16 = 1 , 26 = 64 and 36 = 729 (need two digit number).So, 64 is the number.⇒ 82 = 64 = 43Hence, 64 is the two-digit number which is a square number and also cubic number.

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