TITLE: How to show this fraction is equal to 1/2?
QUESTION [12 upvotes]: I have the fraction:
$$\frac{\left(2 \left(\frac {a}{\sqrt{2}}\right) + a \right) a} {2(1 + \sqrt{2})a^2}$$
Using Mathematica, I've found that this simplifies to $\frac{1}{2}$, but how did it achieve the result? How can I simplify that fraction to $\frac12$?

REPLY [8 votes]: I think you may have missed that by definition,
$\sqrt{2}\sqrt{2}=2$
And thus,
$\frac{2}{\sqrt{2}}=\sqrt{2}$
This simplification issue is quite common. Of course using this and multiplying out/ factoring terms may get your desired result:
$$=\frac{(\sqrt{2}+1)a^2}{2(\sqrt{2}+1)a^2}$$
In which the $\frac{(\sqrt{2}+1)a^2}{(\sqrt{2}+1)a^2}$ reduces to one in the case $a \neq 0$ .