Find mistake in simplifying rational expression

Question

Why can't I simplify the following expression? $$\frac{-(-\frac{\sqrt2}{2})}{(-\frac{\sqrt2}{2})^2}$$ I did this: $$\frac{-(-\frac{\sqrt2}{2})}{(-\frac{\sqrt2}{2})^2}=\frac{-1(-\frac{\sqrt2}{2})}{(-\frac{\sqrt2}{2})^2}=\frac{-1}{(-\frac{\sqrt2}{2})}=\frac{2}{\sqrt2}$$ However, the correct way to do is first raise a square and then simplify: $$\frac{-(-\frac{\sqrt2}{2})}{(-\frac{\sqrt2}{2})^2} = \frac{-(-\frac{\sqrt2}{2})}{\frac{2}{4}}=\frac{\sqrt2}{2}\cdot\frac{4}{2}=\sqrt2$$

Answer 1

There is nothing wrong, in the sense that both final expressions are equivalent: $$\frac2{\sqrt2}=\frac{\sqrt2\cdot\sqrt2}{\sqrt2}=\sqrt2$$ Yet $\sqrt2$ is simpler, both in number of operations and the lack of an irrational denominator.

Answer 2

You are dividing two positive numbers so the result is positive.

Upon simplification of $$ \frac{-(-\frac{\sqrt2}{2})}{(-\frac{\sqrt2}{2})^2}$$ we come up with $$ \frac {1}{ \frac {\sqrt 2}{2}} =\frac {2}{ \sqrt 2}=\sqrt 2$$