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Cube Root Through Prime Factorisation Method

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Cube Root Through Prime Factorisation Method:

You will find that each prime factor in the prime factorisation of the cube of a number, occurs thrice the number of times it occurs in the prime factorisation of the number itself.

Consider 3375.

We find its cube root by prime factorisation:

3375 = 3 × 3 × 3 × 5 × 5 × 5 = 33 × 53 = (3 × 5)3.

Therefore, cube root of 3375 = `root(3)(3375)` = 3 × 5 = 15.
 
Similarly, to find `root(3)(74088)`, we have,
 
74088 = 2 × 2 × 2 × 3 × 3 × 3 × 7 × 7 × 7 = 23 × 33 × 73 = (2 × 3 × 7)3
 
Therefore, `root(3)(74088)` = 2 × 3 × 7 = 42.

Example

Find the cube root of 8000.

Prime factorisation of 8000 is 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5
So, `root(3)(8000) = 2 xx 2 xx 5 = 20`.

Example

Find the cube root of 13824 by prime factorisation method.

13824 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
= 23 × 23 × 23 × 33.
Therefore, `root(3)(13824) = 2 xx 2 xx 2 xx 3 = 24`.

Example

Find the cube root of 216.

216 = 2 × 2 × 2 × 3 × 3 × 3

216 = (3 × 2) × (3 × 2) × (3 × 2) = (3 × 2)3 = 63.

∴ `root(3)(216) = 6 "that is" (216)^(1/3) = 6.`

Example

Find the cube root of -1331.

1331 = 11 × 11 × 11 = 113

-1331 = (-11) × (-11) × (-11) = (-11)3.

∴ `root(3)(-1331) = -11`

Example

Find the cube root of 1728.

1728 = 8 × 216 = 2 × 2 × 2 × 6 × 6 × 6.

∴ 1728 = 23 × 63 = (2 × 6)3 ....[am × bm = (a × b)m]

`root(3)(1728)` = 2 × 6 = 12      ....(Note that, cube root of -1728 is -12.)

Example

Find `root(3)(0.125)`.

`root(3)(0.125)`

`= root(3)(125/1000)`

`= (root(3)(125))/(root(3)(1000))   ....[(a/b)^m = (a^m)/(b^m)]`

`= (root(3)(5^3))/(root(3)(10^3))`

`= 5/10`

= 0.5                .....`(a^m)^(1/m) = a`

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