How to solve $y = \sqrt{k\sqrt{k\sqrt{k \sqrt{k\sqrt{\dots}}}}}$?

Question

$\text{Given}\quad y = \sqrt{k\sqrt{k\sqrt{k \sqrt{k\sqrt{\dots}}}}}\quad \text{ where }\,k\geq 0,\;\;\text{find the value of }\,y.$
I have no idea on how to solve problems like this.

Answer 1

We know that $$ y=\sqrt{k\cdot\sqrt{k\cdot\sqrt{\dots}}} $$ We can square the equation and divide by $k$: $$ \frac{y^2}k=\sqrt{k\cdot\sqrt{\dots}} $$ But the right hand side is just $y$ again, so we have $$ \frac{y^2}k=y $$ Solving this gives $y=0$ or $y=k$.

Answer 2

$$y=\sqrt{k\cdot\sqrt{k\cdot\sqrt{\dots}}}=k^{1/2+1/4+1/8+...}=k^{\frac{1}{2}\frac{1}{1-1/2}}=k^1=k$$