Alebgraic confusion with Square Root

Question

If I define

$y = \frac{\sqrt{g}}{3}$

and $g = x^2$,

does $y$ then become:

$y = \frac{x}{3}$, or

$y = \frac{|x|}{3}$.

I thought the first option is valid, since shouldn't the square root cancel out the square?

Answer 1

Try it with $x=-5$. When you square $x$ you get a positive value. $\sqrt g$ is defined to be positive, so you need the absolute value bars.

Answer 2

Notice that squaring a negative number yields a positive one. So however you slice it, $\sqrt{x^2}$ returns a positive value. Take any negative number to convince yourself.