The fourth power of $\sqrt{1+\sqrt{1+\sqrt{1}}}$ is
$\text{(A) } \sqrt{2}+\sqrt{3}\quad \text{(B) } \tfrac{1}{2}(7+3\sqrt{5})\quad \text{(C) } 1+2\sqrt{3}\quad \text{(D) } 3\quad \text{(E) } 3+2\sqrt{2}$
We can simplify the expression to $\sqrt{1+\sqrt{2}}$ Then we square it so it is now $1+\sqrt{2}$. We still have to square it one more time, getting $\boxed{(E)\ 3+2\sqrt{2}}$.