# Indices And Cube Root

##### Class 8th Mathematics (new) MHB Solution

**Practice Set 3.1**

- Express the following numbers in index form.(1) Fifth root of 13(2) Sixth root of 9(3)…
- Write in the form ‘nth root of a’ in each of the following numbers.1. (81)1/4 2.…

**Practice Set 3.2**- Complete the following table.
- Write the following number in the form of rational indices.(1) Square root of 5th power…

**Practice Set 3.3**- Find the cube root of the following numbers.8000
- Find the cube root of the following numbers.729
- Find the cube root of the following numbers.343
- Find the cube root of the following numbers.-512
- Find the cube root of the following numbers.-2744
- Find the cube root of the following numbers.32768
- Simplify:(1) cube root { {27}/{125} } (2) cube root { {16}/{54} } (3) If…

- Express the following numbers in index form.(1) Fifth root of 13(2) Sixth root of 9(3)…
- Write in the form ‘nth root of a’ in each of the following numbers.1. (81)1/4 2.…

**Practice Set 3.2**

- Complete the following table.
- Write the following number in the form of rational indices.(1) Square root of 5th power…

**Practice Set 3.3**

- Find the cube root of the following numbers.8000
- Find the cube root of the following numbers.729
- Find the cube root of the following numbers.343
- Find the cube root of the following numbers.-512
- Find the cube root of the following numbers.-2744
- Find the cube root of the following numbers.32768
- Simplify:(1) cube root { {27}/{125} } (2) cube root { {16}/{54} } (3) If…

###### Practice Set 3.1

**Question 1.**Express the following numbers in index form.

(1) Fifth root of 13

(2) Sixth root of 9

(3) Square root of 256

(4) Cube root of 17

(5) Eighth root of 100

(6) Seventh root of 30

**Answer:**(1) Fifth root of 13

In general, n^{th} root of ‘a’ is expressed as .

So, the fifth root of 13 is expressed as .

Here, 13 is base, is the index and is the index form of the number.

(2) Sixth root of 9

In general, n^{th} root of ‘a’ is expressed as .

So, the sixth root of 9 is expressed as .

Here, 9 is base, is the index and is the index form of the number.

(3) Square root of 256

In general, n^{th} root of ‘a’ is expressed as .

So, the square root of 256 is expressed as .

Here, 256 is base, is the index and is the index form of the number.

(4) Cube root of 17

In general, n^{th} root of ‘a’ is expressed as .

So, cube root of 17 is expressed as.

Here, 17 is base, is the index and is the index form of the number.

(5) Eighth root of 100

In general, n^{th} root of ‘a’ is expressed as .

So, the eighth root of 100 is expressed as .

Here, 100 is base, is the index and is the index form of the number.

(6) Seventh root of 30

In general, n^{th} root of ‘a’ is expressed as .

So, the seventh root of 30 is expressed as .

Here, 30 is base, is the index and is the index form of the number.

**Question 2.**Write in the form ‘n^{th} root of a’ in each of the following numbers.

1. (81)^{1/4} 2. (49)^{1/2}

3. (15)^{1/5} 4. (512)^{1/9}

5. (100)^{1/19} 6. (6)^{1/7}

**Answer:**1. (81)^{1/4}

In general, a^{1/n} is written as ‘n^{th} root of a’.

So, (81)^{1/4} is written as ‘4^{th} root of 81’.

2. (49)^{1/2}

In general, a^{1/n} is written as ‘n^{th} root of a’.

So, (49)^{1/2} is written as ‘square root of 49’.

3. (15)^{1/5}

In general, a^{1/n} is written as ‘n^{th} root of a’.

So, (15)^{1/5} is written as ‘5^{th} root of 15’.

4. (512)^{1/9}

In general, a^{1/n} is written as ‘n^{th} root of a’.

So, (512)^{1/9} is written as ‘9^{th} root of 512’.

5. (100)^{1/19}

In general, a^{1/n} is written as ‘n^{th} root of a’.

So, (100)^{1/19} is written as ‘19^{th} root of 100’.

6. (6)^{1/7}

In general, a^{1/n} is written as ‘n^{th} root of a’.

So, (6)^{1/7} is written as ‘7^{th} root of 6’.

**Question 1.**

Express the following numbers in index form.

(1) Fifth root of 13

(2) Sixth root of 9

(3) Square root of 256

(4) Cube root of 17

(5) Eighth root of 100

(6) Seventh root of 30

**Answer:**

(1) Fifth root of 13

In general, n^{th} root of ‘a’ is expressed as .

So, the fifth root of 13 is expressed as .

Here, 13 is base, is the index and is the index form of the number.

(2) Sixth root of 9

In general, n^{th} root of ‘a’ is expressed as .

So, the sixth root of 9 is expressed as .

Here, 9 is base, is the index and is the index form of the number.

(3) Square root of 256

In general, n^{th} root of ‘a’ is expressed as .

So, the square root of 256 is expressed as .

Here, 256 is base, is the index and is the index form of the number.

(4) Cube root of 17

In general, n^{th} root of ‘a’ is expressed as .

So, cube root of 17 is expressed as.

Here, 17 is base, is the index and is the index form of the number.

(5) Eighth root of 100

In general, n^{th} root of ‘a’ is expressed as .

So, the eighth root of 100 is expressed as .

Here, 100 is base, is the index and is the index form of the number.

(6) Seventh root of 30

In general, n^{th} root of ‘a’ is expressed as .

So, the seventh root of 30 is expressed as .

Here, 30 is base, is the index and is the index form of the number.

**Question 2.**

Write in the form ‘n^{th} root of a’ in each of the following numbers.

1. (81)^{1/4} 2. (49)^{1/2}

3. (15)^{1/5} 4. (512)^{1/9}

5. (100)^{1/19} 6. (6)^{1/7}

**Answer:**

1. (81)^{1/4}

In general, a^{1/n} is written as ‘n^{th} root of a’.

So, (81)^{1/4} is written as ‘4^{th} root of 81’.

2. (49)^{1/2}

In general, a^{1/n} is written as ‘n^{th} root of a’.

So, (49)^{1/2} is written as ‘square root of 49’.

3. (15)^{1/5}

In general, a^{1/n} is written as ‘n^{th} root of a’.

So, (15)^{1/5} is written as ‘5^{th} root of 15’.

4. (512)^{1/9}

In general, a^{1/n} is written as ‘n^{th} root of a’.

So, (512)^{1/9} is written as ‘9^{th} root of 512’.

5. (100)^{1/19}

In general, a^{1/n} is written as ‘n^{th} root of a’.

So, (100)^{1/19} is written as ‘19^{th} root of 100’.

6. (6)^{1/7}

In general, a^{1/n} is written as ‘n^{th} root of a’.

So, (6)^{1/7} is written as ‘7^{th} root of 6’.

###### Practice Set 3.2

**Question 1.**Complete the following table.

**Answer:**

Explanation of Table

Generally we can express two meaning of the number a^{m/n}.

a^{m/n} = (a^{m})^{1/n} means ‘n^{th} root of m^{th} powerof a’.

a^{m/n} = ()^{m} means ‘m^{th} power of n^{th} root of a’.

(1) (225)^{3/2}

(225^{3})^{1/2} means ‘Cube of square root of 225’.

(225^{1/2})^{3} means ‘Square root of cube of 225’.

(2) (45)^{4/5}

(45^{4})^{1/5} means ‘Fourth power of fifth root of 45’.

(45^{1/5})^{4} means ‘Fifth root of fourth power of 45’.

(3) (81)^{6/7}

(81^{6})^{1/7} means ‘Sixth power of seventh root of 81’.

(81^{1/7})^{6} means ‘Seventh root of sixth power of 81’.

(4) (100)^{4/10}

(100^{4})^{1/10} means ‘Fourth power of tenth root of 100’.

(100^{1/10})^{4} means ‘Tenth root of fourth power of 100’.

(5) (21)^{3/7}

(21^{3})^{1/7} means ‘Cube of seventh root of 21’.

()^{3} means ‘Seventh root of cube of 21’.

**Question 2.**Write the following number in the form of rational indices.

(1) Square root of 5^{th} power of 121.

(2) Cube of 4^{th} root of 324.

(3) 5^{th} root of square of 264.

(4) Cube of cube root of 3.

**Answer:**we know that ‘n^{th} root of m^{th} powerof a’ is expressed as (a^{m})^{1/n}

and ‘m^{th} power of n^{th} root of a’ is expressed as ()^{m}.

(1) Square root of 5^{th} power of 121.

We know that,

‘n^{th} root of m^{th} powerof a’ is expressed as (a^{m})^{1/n}

So, ‘Square root of 5^{th} power of 121’ is expressed as (121^{5})^{1/2} or (121)^{5/2}.

(2) Cube of 4^{th} root of 324.

We know that,

‘n^{th} root of m^{th} powerof a’ is expressed as (a^{m})^{1/n}

So, ‘Cube of 4^{th} root of 324’ is written as (324^{1/4})^{3} or (324)^{3/4}.

(3) 5^{th} root of square of 264.

We know that,

‘n^{th} root of m^{th} powerof a’ is expressed as (a^{m})^{1/n}

So, ‘5^{th} root of square of 264’ is written as (264^{2})^{1/5} or

(264)^{2/5}.

(4) Cube of cube root of 3.

We know that,

‘m^{th} power of n^{th} root of a’ is expressed as ()^{m}

So, ‘Cube of cube root of 3’ is written as (3^{1/3})^{3} or (31)^{3/3}.

**Question 1.**

Complete the following table.

**Answer:**

Explanation of Table

Generally we can express two meaning of the number a^{m/n}.

a^{m/n} = (a^{m})^{1/n} means ‘n^{th} root of m^{th} powerof a’.

a^{m/n} = ()^{m} means ‘m^{th} power of n^{th} root of a’.

(1) (225)^{3/2}

(225^{3})^{1/2} means ‘Cube of square root of 225’.

(225^{1/2})^{3} means ‘Square root of cube of 225’.

(2) (45)^{4/5}

(45^{4})^{1/5} means ‘Fourth power of fifth root of 45’.

(45^{1/5})^{4} means ‘Fifth root of fourth power of 45’.

(3) (81)^{6/7}

(81^{6})^{1/7} means ‘Sixth power of seventh root of 81’.

(81^{1/7})^{6} means ‘Seventh root of sixth power of 81’.

(4) (100)^{4/10}

(100^{4})^{1/10} means ‘Fourth power of tenth root of 100’.

(100^{1/10})^{4} means ‘Tenth root of fourth power of 100’.

(5) (21)^{3/7}

(21^{3})^{1/7} means ‘Cube of seventh root of 21’.

()^{3} means ‘Seventh root of cube of 21’.

**Question 2.**

Write the following number in the form of rational indices.

(1) Square root of 5^{th} power of 121.

(2) Cube of 4^{th} root of 324.

(3) 5^{th} root of square of 264.

(4) Cube of cube root of 3.

**Answer:**

we know that ‘n^{th} root of m^{th} powerof a’ is expressed as (a^{m})^{1/n}

and ‘m^{th} power of n^{th} root of a’ is expressed as ()^{m}.

(1) Square root of 5^{th} power of 121.

We know that,

‘n^{th} root of m^{th} powerof a’ is expressed as (a^{m})^{1/n}

So, ‘Square root of 5^{th} power of 121’ is expressed as (121^{5})^{1/2} or (121)^{5/2}.

(2) Cube of 4^{th} root of 324.

We know that,

‘n^{th} root of m^{th} powerof a’ is expressed as (a^{m})^{1/n}

So, ‘Cube of 4^{th} root of 324’ is written as (324^{1/4})^{3} or (324)^{3/4}.

(3) 5^{th} root of square of 264.

We know that,

‘n^{th} root of m^{th} powerof a’ is expressed as (a^{m})^{1/n}

So, ‘5^{th} root of square of 264’ is written as (264^{2})^{1/5} or

(264)^{2/5}.

(4) Cube of cube root of 3.

We know that,

‘m^{th} power of n^{th} root of a’ is expressed as ()^{m}

So, ‘Cube of cube root of 3’ is written as (3^{1/3})^{3} or (31)^{3/3}.

###### Practice Set 3.3

**Question 1.**Find the cube root of the following numbers.

8000

**Answer:**First find the factor of 8000

8000 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5

For finding the cube root, we pair the prime factors in 3’s.

= (2 × 2 × 5)^{3}

= (2 × 10)^{3}

= 20^{3}

i.e. cube root of 8000 = (8000)^{1/3} = (20^{3})^{1/3} = 20 (answer).

**Question 2.**Find the cube root of the following numbers.

729

**Answer:**First find factors of 729

729 = 9 × 9 × 9

For finding the cube root, we pair the prime factors in 3’s.

= 9^{3}

i.e. cube root of 729 = (729)^{1/3} = (9^{3})^{1/3} = 9 (answer).

**Question 3.**Find the cube root of the following numbers.

343

**Answer:**First find the factor of 343

343 = 7 × 7 × 7

For finding the cube root, we pair the prime factors in 3’s.

= 7^{3}

i.e. cube root of 343 = (343)^{1/3} = (7^{3})^{1/3} = 7 (answer).

**Question 4.**Find the cube root of the following numbers.

-512

**Answer:**First find factors of - 512

-512 = (-8) × (-8) × (-8)

For finding the cube root, we pair the prime factors in 3’s.

= (-8)^{3}

i.e. cube root of -512 = (- 512)^{1/3} = (-8^{3})^{1/3} = -8 (answer).

**Question 5.**Find the cube root of the following numbers.

-2744

**Answer:**First find factors of -2744

-2744 = (-14) × (-14) × (-14)

For finding the cube root, we pair the prime factors in 3’s.

= (-14)^{3}

i.e. cube root of -2744 = (- 2744)^{1/3} = (-14^{3})^{1/3} = -14 (answer).

**Question 6.**Find the cube root of the following numbers.

32768

**Answer:**First find factor of 32768

32768 = 32 × 32 × 32

For finding the cube root, we pair the prime factors in 3’s.

= 32^{3}

i.e. cube root of 32768 = ∛32768 = (32^{3})^{1/3} = 32 (answer).

**Question 7.**Simplify:

(1)

(2)

(3) If = 9 then =

**Answer:**(1)

(answer).

(2)

(answer).

3) If ∛729 = 9 then ∛0.000729 = ?

We know that ∛729 = 9

S0 (answer).

**Question 1.**

Find the cube root of the following numbers.

8000

**Answer:**

First find the factor of 8000

8000 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5

For finding the cube root, we pair the prime factors in 3’s.

= (2 × 2 × 5)^{3}

= (2 × 10)^{3}

= 20^{3}

i.e. cube root of 8000 = (8000)^{1/3} = (20^{3})^{1/3} = 20 (answer).

**Question 2.**

Find the cube root of the following numbers.

729

**Answer:**

First find factors of 729

729 = 9 × 9 × 9

For finding the cube root, we pair the prime factors in 3’s.

= 9^{3}

i.e. cube root of 729 = (729)^{1/3} = (9^{3})^{1/3} = 9 (answer).

**Question 3.**

Find the cube root of the following numbers.

343

**Answer:**

First find the factor of 343

343 = 7 × 7 × 7

For finding the cube root, we pair the prime factors in 3’s.

= 7^{3}

i.e. cube root of 343 = (343)^{1/3} = (7^{3})^{1/3} = 7 (answer).

**Question 4.**

Find the cube root of the following numbers.

-512

**Answer:**

First find factors of - 512

-512 = (-8) × (-8) × (-8)

For finding the cube root, we pair the prime factors in 3’s.

= (-8)^{3}

i.e. cube root of -512 = (- 512)^{1/3} = (-8^{3})^{1/3} = -8 (answer).

**Question 5.**

Find the cube root of the following numbers.

-2744

**Answer:**

First find factors of -2744

-2744 = (-14) × (-14) × (-14)

For finding the cube root, we pair the prime factors in 3’s.

= (-14)^{3}

i.e. cube root of -2744 = (- 2744)^{1/3} = (-14^{3})^{1/3} = -14 (answer).

**Question 6.**

Find the cube root of the following numbers.

32768

**Answer:**

First find factor of 32768

32768 = 32 × 32 × 32

For finding the cube root, we pair the prime factors in 3’s.

= 32^{3}

i.e. cube root of 32768 = ∛32768 = (32^{3})^{1/3} = 32 (answer).

**Question 7.**

Simplify:

(1)

(2)

(3) If = 9 then =

**Answer:**

(1)

(answer).

(2)

(answer).

3) If ∛729 = 9 then ∛0.000729 = ?

We know that ∛729 = 9

S0 (answer).