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?
 A: Base case: $\sqrt{k}=k^{1/2}=k^{1-1/2^1}$.
Induction step: let $a_n$ be the LHS when there are $n$ square root signs. If $a_n=k^{1-1/2^n}$, then
$$
a^2_{n+1}=ka_n=kk^{1-1/2^n}=k^{2-1/2^n}\implies a_{n+1}=k^{1-1/2^{n+1}}.
$$
A: $$\underbrace{\sqrt{k\sqrt{k\sqrt{k\sqrt{\cdots\sqrt{k}}}}}}_{n\text { times}} = \sqrt{k}\times\sqrt[4]{k}\times \ldots \times \sqrt[2^n]{k} = k^{1/2}\times k^{1/4}\times \ldots \times k^{1/2^n} = \\ = k^{\Large\frac{1}{2} + \frac{1}{4} + \ldots + \frac{1}{2^n}} = k^{\Large1-\frac{1}{2^n}}$$
A: Take the logarithm (as already suggested in comments):
$$ \frac{1}{2}\left(\ln k +  \frac{1}{2}\left(\ln k + \frac{1}{2}\left( \ln k + \ldots \right) \right)\right)$$
$$ =\left(\frac{1}{2}+ \frac{1}{2^2} + \frac{1}{2^3} + \ldots + \frac{1}{2^n}\right)\ln k$$
$$ =\left(1 - \frac{1}{2^n}\right) \ln k.$$
A: The first $k$ is powered by $\frac{1}{2}$, the second by $\frac{1}{2^2}$ the $n$-therm is then $k^{\frac{1}{2^n}}$ then the expression is equal to:
$$k^{\frac{1}{2^1}}\cdot k^{\frac{1}{2^2}}\cdot \dots  k^{\frac{1}{2^n}}=k^{\sum_{i=1}^n \frac{1}{2^i}}$$
