Question

Evaluate the following:

`(-32)^(2/5)`

Answer 1

Given: `(-32)^(2/5)`

Step-wise calculation:

1. Express the base (–32) as a power of (–2):

–32 = –(2⁵)

However, since the exponent is a fractional power with denominator 5, and the root of degree 5 is an odd root, the real fifth root of a negative number exists.

2. Compute the fifth root first, the inner root corresponding to the denominator 5 in the exponent: 

`(-32)^(1//5) = -2`   ...(∵ (–2)⁵ = –32)

3. Now, apply the square power corresponding to numerator 2:

`((-32)^(1//5))^2 = (-2)^2`

`((-32)^(1//5))^2 = 4`