\[ \begin{split} &\lim_{n \to \infty} \sqrt{2}\cdot \sqrt[4]{2}\cdot \sqrt[8]{2}\cdot ...\cdot \sqrt[2^n]{2}=\\[16pt] &=\lim_{n \to \infty} 2^\tfrac{1}{2}\cdot 2^\tfrac{1}{4}\cdot ...\cdot 2^\tfrac{1}{2^n}=\\[16pt] &=\lim_{n \to \infty} 2^{\tfrac{1}{2}+\tfrac{1}{4}+\tfrac{1}{8}+...+\tfrac{1}{2^n}}=\\[16pt] &= 2^{\tfrac{\tfrac{1}{2}}{1-\tfrac{1}{2}}}=\\[16pt] &= 2^{\tfrac{1}{2}\cdot \tfrac{2}{1}}=\\[16pt] &= 2^1=2 \end{split} \]