How to simplify this expression of square roots? If one simplifies the infinite nested radical
$$\sqrt{x+\sqrt{x^2+\sqrt{x^4+\sqrt{x^8+\dots}}}}$$
Does one get $$\sqrt x\frac{1+\sqrt5}2$$
Any help would be appreciated!

You have 
$$\sqrt{x+\sqrt{x^2+\sqrt{x^4+\sqrt{x^8+\dots}}}}$$
$$ = \sqrt{x+x\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}$$
$$ = \sqrt{x}\sqrt{1+1\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}$$
The continued square root $\sqrt{1+1\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}$ is a well known form of $\phi = \frac{\sqrt{5}+1}{2}$ (see here) and so you end up with:
$$ \frac{\sqrt{x}(\sqrt{5}+1)}{2}$$
as required.
