<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] Yes. $$\frac{x^a}{x^b} = x^{a-b}$$ In this case we have $$\frac{2^1}{\sqrt{2}} = \frac{2^1}{2^{1/2}} = 2^{1- 1/2} = 2^{1/2} = \sqrt{2}$$
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