TITLE: is there a general formula for cases like $\sqrt{2}$ = $\frac {2}{\sqrt{2}}$?
QUESTION [6 upvotes]: I just noticed that $\sqrt{2}$ is equal to $\frac {2}{\sqrt{2}}$:
$\sqrt{2} = 1.414213562$
$\frac {2}{\sqrt{2}} = 1.414213562$
It is confirmed by a hand-calculator.
I tried to proof this as follows:
$\sqrt{2}$ = $\frac {2}{\sqrt{2}}$
$2^{\frac 12} = \frac {2}{2^{\frac 12}}$
$2 = 2^{\frac 12} \cdot 2^{\frac 12}$
$2 = 2^{\frac 12 + \frac 12}  $
$ 2 = 2^1 $
$ 2 = 2$
It is also true for:
$-\frac {2}{\sqrt5} = \frac {-2}{5}\sqrt 5$
I didn't know this relationship beforehand and it was new for me, my question is: Is there a general rule for this?

REPLY [5 votes]: It's more fundamental than the formula $x^a/x^b=x^{a-b}$. The definition of square root is $\sqrt{a}\sqrt{a}=a$. Just divide both sides by $\sqrt{a}$.