Indices And Cube Root

(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.

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’.

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

(225³)^1/2 means ‘Cube of square root of 225’.

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

(2) (45)^4/5

(45⁴)^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

(81⁶)^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

(100⁴)^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

(21³)^1/7 means ‘Cube of seventh root of 21’.

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

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 (121⁵)^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 (264²)^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.

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

= 20³

i.e. cube root of 8000 = (8000)^1/3 = (20³)^1/3 = 20 (answer).

First find factors of 729

729 = 9 × 9 × 9

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

= 9³

i.e. cube root of 729 = (729)^1/3 = (9³)^1/3 = 9 (answer).

First find the factor of 343

343 = 7 × 7 × 7

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

= 7³

i.e. cube root of 343 = (343)^1/3 = (7³)^1/3 = 7 (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 = (-8³)^1/3 = -8 (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 = (-14³)^1/3 = -14 (answer).

First find factor of 32768

32768 = 32 × 32 × 32

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

= 32³

i.e. cube root of 32768 = ∛32768 = (32³)^1/3 = 32 (answer).

(1)

(answer).

(2)

(answer).

3) If ∛729 = 9 then ∛0.000729 = ?

We know that ∛729 = 9

S0 (answer).