Simplify.

(–32)^0.6

1 answer

To simplify \((-32)^{0.6}\), we first note that raising a negative number to a non-integer exponent can result in a complex number. The exponent \(0.6\) can be expressed as a fraction:

\[ 0.6 = \frac{6}{10} = \frac{3}{5} \]

Thus, we can rewrite the expression as:

\[ (-32)^{0.6} = (-32)^{\frac{3}{5}} = \left((-32)^{3}\right)^{\frac{1}{5}} \]

Calculating \((-32)^3\):

\[ (-32)^3 = -32768 \]

Next, we need to find the fifth root of \(-32768\):

\[ (-32768)^{\frac{1}{5}} \]

We can express \(-32768\) in terms of powers:

\[ -32768 = -32^5 \]

So:

\[ (-32768)^{\frac{1}{5}} = (-32^5)^{\frac{1}{5}} = -32 \]

Thus, we find that:

\[ (-32)^{0.6} = -32 \]

Therefore, the simplified result is:

\[ \boxed{-32} \]