How do I simplify the square root of a negative number squares?

Question

How would I simplify $\sqrt{h^2}$, where $h$ is a negative number? The answer key says $-h$ but I don't understand the answer.

Answer 1

Note that if $h$ is negative, then $-h$ is positive and $h^2=(-h)^2$. Thus $$ \sqrt{h^2} = \sqrt{(-h)^2} = -h, $$ where the second equality follows because $\sqrt{p^2}=p$ whenever $p$ is positive.

In general, we have $\sqrt{x^2}=|x|$, where $$ |x| := \begin{cases} x & \text{if}\ x\geq 0 \\ -x & \text{if}\ x<0. \end{cases} $$