TITLE: Raising both sides of an equation to the same negative power
QUESTION [9 upvotes]: If I raise both sides of an equation to the same negative exponent will the equality remain?
$$a = b$$

REPLY [3 votes]: Given $$a=b$$ you want to know if $$a^n = b^n$$ for some negative value of $n.$  Since $a=b$, let's replace them both with $c$ in the first equation: $$c=c$$ Now let's put both sides of the equation to the power of $n$: $$c^n=c^n$$ Finally, let's replace the first $c$ with $a$ (because $a=c$) and the second $c$ with $b$ (same reasoning​): $$a^n=b^n$$ Looks good.