Question

What is th value of √(2 √(2 √(2 √(2)))) ?

Answer 1

$$\begin{gathered} \sqrt{\left(2\sqrt{\left(2\sqrt{\left(2\sqrt{\left(2\right)}\right)}\right)}\right)} \\ =\sqrt{\left(2\sqrt{\left(2\sqrt{\left(2^1\times 2^{\frac{1}{2}}\right)}\right)}\right)} \\ =\sqrt{\left(2\sqrt{\left(2\sqrt{\left(2^{\frac{3}{2}}\right)}\right)}\right)} \\ =\sqrt{\left(2\sqrt{\left(2\times 2^{\frac{3}{4}}\right)}\right)} \\ =\sqrt{\left(2\sqrt{\left(2^{\frac{7}{4}}\right)}\right)} \\ =\sqrt{\left(2\times 2^{\frac{7}{8}}\right)} \\ =\sqrt{\left(2^{\frac{15}{8}}\right)} \\ =2^{\frac{15}{16}} \end{gathered}$$