<s>[INST] is there a general formula for cases like $\sqrt{2}$ = $\frac {2}{\sqrt{2}}$? 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?
 [/INST] It will be true for any positive real $x$ that
$$\frac{x}{\sqrt{x}} = \frac{x}{x^{1/2}} = x^1\cdot x^{-1/2} = x^{1-1/2} = x^{1/2} = \sqrt{x}$$
</s>