I am overthinking this but (−4)(−52) is not a valid equation, am I correct? Working...

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 ${\displaystyle -\frac{1}{32}i}$, where ${\displaystyle i}$ is called the Imaginary Unit, and is equal to ${\displaystyle \sqrt{-1}}$

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