10. $[8]$ Compute
$$
\prod_{n=0}^{\infty}\left(1-\left(\frac{1}{2}\right)^{3^{n}}+\left(\frac{1}{4}\right)^{3^{n}}\right)
$$
Answer: $\frac{2}{3}$ We can rewrite each term as $\frac{1+\left(\frac{1}{2}\right)^{3^{n+1}}}{1+\left(\frac{1}{2}\right)^{3^{n}}}$. In the infinite product, each term of the form $1+\left(\frac{1}{2}\right)^{3^{n}}$ with $n>0$ appears once in the numerator and once in the denominator. The only remaining term is $1+\left(\frac{1}{2}\right)^{1}$ in the first denominator.