Raising a negative number to an odd negative fractional exponent

Question

Perhaps I am overthinking this but $(-4)^{(-5/2)}$ is not a valid equation, am I correct? Working through the problem gives me $1/(-4^{5/2})$ which then works out to $1/\sqrt{-4^5}$ which leaves a negative number in the square root, which is not valid.

Answer 1

It is perfectly valid, the result is simply not a Real Number, it is what is called an Imaginary Number (or a Complex Number in the case in which you had a Real Numbered component as well), which you may not be familiar with.

The result you gave simplifies down to $-\frac{1}{32}i$, where $i$ is called the Imaginary Unit, and is equal to $\sqrt{-1}$.

If you cared to investigate this further try reading these for a decent introduction: Complex Numbers on Wikipedia and Imaginary Numbers. Khan Academy also has a good video on this subject for the introductory level.

Answer 2

The square root of a negative number is a complex number. $(-4)^{-{5/2}} = {1 \over (\sqrt{-4})^5} = {1 \over (2i)^5}=-\frac{1}{32} i$. If this sounds like complete gibberish to you, google "imaginary numbers" and read up on it.