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We have learnt that the numbers 8, 27, 64, 125, etc. are perfect cubes. Now, look at the following
expressions:

3
3
• 8 = 2 × 2 × 2 = 2 64 = 4 × 4 × 4 = 4 3 27 = 3 × 3 × 3 = 3 125 = 5 × 5 × 5 = 5 3
We can say that, finding the cube root is the inverse operation of finding the cube of a number.

8 is the cube of 2 ⇒ 2 is the cube root of 8 ( 8 = 2).
3
Note: For any number n, n
3
3
27 is the cube of 3 ⇒ 3 is the cube root of 27 ( 27 = 3). is called a radical, n is
called a radicand and
64 is the cube of 4 ⇒ 4 is the cube root of 64 ( 64 = 4). 3 is called the index of
3
3
125 is the cube of 5 ⇒ 5 is the cube root of 125 ( 125 = 5). the radical.
In general, we can say that,

3
3
A number ‘m’ is the cube root of a number ‘n’, if n = m . That is m = n
Finding the Cube Root of a Number

We can find the cube root of a number by the following methods:

Prime Factorisation Method

To find the cube root using the prime factorisation, follow the given steps.

Steps Example: Find the cube root of 2744
1. Find the prime factorisation of the given number. 2744 = 2 × 2 × 2 × 7 × 7 × 7
2. Group the factors in triplets such that all three factors in each 2744 = 2 × 2 × 2 × 7 × 7 × 7
triplet are the same.
3. Find the product of prime factors by taking one factor from ∴ 3 2744 = 2 × 7 = 14
each triplet. The product obtained is the cube root of the given
number.
Example 10: Find the cube root of each of the following numbers by using the prime factorisation
method.

(a) 175616 (b) –13824 2 175616
2 87808
Solution: (a) Prime factorisation of 175616 2 43904
= 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 7 × 7 × 7 2 21952 2 13824
2 10976 2 6912
3
3
3
3
= 2 × 2 × 2 × 7 = (2 × 2 × 2 × 7) 3 2 5488 2 3456
2 2744 2 1728
∴ 3 175616 = (2 × 2 × 2 × 7) = 56 2 1372 2 864
(b) Prime factorisation of 13824 2 686 2 432
7 343 2 216
= (2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3) 7 49 2 108
7 7 2 54
∴ –13824 = – (2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3) 1 3 27
3
3
3
3
= – (2 × 2 × 2 × 3 ) = –(2 × 2 × 2 × 3) 3 3 9
3 3
∴ −(13824 = –(2 × 2 × 2 × 3) = –24 1
)
3
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