Q:

What is the reciprocal of $(-1/2)^k$?

What is the reciprocal of $(-1/2)^k$?
The answer is meant to be $2^{-k}$ as if you flip something upside down the power becomes negative. However, I am not sure what happens to the negative in front of the fraction. 

A:

The reciprocal of $x$ is simply $\frac1x$. So the reciprocal of $(-\tfrac12)^k$ is $\frac1{(-\tfrac12)^k}$. 
You may also recall from the exponentiation laws that $(\frac1x)^y=x^{-y}$, and that the repciprocal of $x^y$ is $\frac1{x^y}=x^{-y}$.
This allows us to rewrite $(-\tfrac 12)^k$ as $(-2)^{-k}$, and then the reciprocal of this is $(-2)^k$. If $k$ is even, this simplifies to $2^k$; if $k$ is odd, it becomes $-2^k$.

