General formula of repeated roots. Prove that $$\underbrace{\sqrt{k\sqrt{k\sqrt{k\sqrt{\cdots\sqrt{k}}}}}}_{n\text { times}}=k^{1-1/2^n}$$
How do I derive this formula?

Take the logarithm (as already suggested in comments):
$$ \frac{1}{2}\left(\ln k +  \frac{1}{2}\left(\ln k + \frac{1}{2}\left( \ln k + \ldots \right) \right)\right)$$
$$ =\left(\frac{1}{2}+ \frac{1}{2^2} + \frac{1}{2^3} + \ldots + \frac{1}{2^n}\right)\ln k$$
$$ =\left(1 - \frac{1}{2^n}\right) \ln k.$$
