# Cube Roots

If the cube of 2 is 8, then 2 is said to be the cube root of 8. We denote cube root by the symbol

Thus

Cube root : The cube root of a given number is that number whose cube is equal to the given number.

Cube Root of a Perfect Cube by Prime Factorisation

We have seen that a number is a perfect cube if it can be expressed as the product of triples of equal factors.

The following example shows how this can be used to find the cube root of a perfect cube.

Find the cube root of 1728.

1. Carry out the prime factorisation of the given number.

1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3

2. Make triples of equal factors.

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

3. Take one prime factor from each triple and find their product.

2 × 2 × 3 = 12

12 is the required cube root, i.e.** **

Find the cube root of 729

729 = 3 × 3 × 3 × 3 × 3 × 3.

729 = 3^{3}× 3^{3 }

Cube Root of a Negative Number :-

We have seen that the cube of a negative integer is negative.

**Example :** (- 5)^{3} = (- 5) x (- 5) x (- 5) = (- 125)

It follows that = (- 5)

**Example :** = - = (- 5)

In general, if x is a positive integer,

Thus, to find the cube root of a negative integer, find the cube root of its absolute value, and put a negative sign before it.

**Example :**

Cube root of a perfect cube without using prime factorization.

We can estimate cube root of given number.

Consider the number 19683.

**1.** Split the number from the right into group of three digit numbers.

19683.

We have two parts 19 and 683.

**2.** The unit digit of 683 is 3. Hence the unit digit of required cube root must be 7.

**3.** We have to estimate the cube root of 19.

8 < 19 < 27.

< <

2 < < 3.

**4.** Take unit digit of the smallest number.

The cube root of 19 lies between 2 and 3. It is â€˜2â€™.

So, the cube root of 19683 is 27.

Estimate the cube root of 110592

**i.**110 592

**ii.**The unit digit of 592 is 2. So, the unit digit of required cube root is 8.

**iii.**Now, 64 < 110 < 125.

< <

4 < < 5

**iv.**The smallest number is 4.

**v.**Therefore, the cube root of 110592 is 48.

For your Knowledge :-

Cube Root of a Perfect Cube Expressed in Exponential Form

Consider the number 13824. In the exponential form it can be written as 13824 = 2^{9}× 3^{3}

If the indices of the factors are multiplies of 3, then the number is a perfect cube. Even if one factor has an index which is not a multiple of 3, then it is not a perfect cube.

**Example** **:** 6912 = 2^{8}× 3^{3} is not a perfect cube.

The cube root is obtained by dividing each index by 3.

Thus

Cube Root of Product of Integers

64 = 2 × 2 × 2 × 2 × 2 × 2

125 = 5 × 5 × 5

Notice also that,

**Cube Root of a Rational Number :-**

We have seen earlier that

It follows from this that

Find the cube root of ** **