$1.81 \sqrt{1+\sqrt{1+\sqrt{1}}}$ to the fourth power is
(A) $\sqrt{2}+\sqrt{3}$.
(B) $\frac{7+3 \sqrt{5}}{2}$.
(C) $1+2 \sqrt{3}$.
(D) 3 .
(E) $3+2 \sqrt{2}$.
(21st American High School Mathematics Examination, 1970)
[Solution] Let $x=\sqrt{1+\sqrt{1+\sqrt{1}}}$, since $\sqrt{1}=1$, then
$$
x=\sqrt{1+\sqrt{2}} \text {. }
$$

Thus, $x^{2}=1+\sqrt{2}$, and $x^{4}=(1+\sqrt{2})^{2}=3+2 \sqrt{2}$. Therefore, the answer is $(E)$.