The value of $(\sqrt{1+\sqrt{1+\sqrt{1}}})^{4}$ is:
(a) $\sqrt{2}+\sqrt{3}$;
(b) $\frac{1}{2}(7+3 \sqrt{5})$;
(c) $1+2 \sqrt{3}$;
(d) 3 ;
(e) $3+2 \sqrt{2}$.
The correct option is (e).

$$
(\sqrt{1+\sqrt{1+\sqrt{1}}})^{4}=(1+\sqrt{2})^{2}=1+2 \sqrt{2}+2=3+2 \sqrt{2}
$$