Square Roots

$\begin{aligned} \sqrt{11 \sqrt{11 \sqrt{11 \sqrt{11}}}} & = \left(11 \left(11 \left(11 \times 11 ^{\frac{1}{2}} \right)^{\frac{1}{2}} \right)^{\frac{1}{2}} \right)^{\frac{1}{2}} \\ & = \left(11 \left(11 \left(11 ^{\frac{3}{2}} \right)^{\frac{1}{2}} \right)^{\frac{1}{2}} \right)^{\frac{1}{2}} \\ & = \left(11 \left(11 \times 11 ^{\frac{3}{4}} \right)^{\frac{1}{2}} \right)^{\frac{1}{2}} \\ & = \left(11 \times 11 ^{\frac{7}{8}} \right)^{\frac{1}{2}} \\ & = \boxed{11 ^{\frac{15}{16}}} \end{aligned}$

The question is in the form of $\sqrt{11\sqrt{11\sqrt{11\sqrt{11}}}}$ , which may be rewritten as $\sqrt{11\sqrt{11\sqrt{11^\frac{3}{2}}}}$ , then $\sqrt{11\sqrt{11({11^\frac{3}{4}}})}$ , then $\sqrt{11(11^\frac{7}{8})}$ , and finally, $11^\frac{15}{16}$

If we solve the expression we get 11^[1/2+1/4+1/8+1/6] solving this GP [1/2(1-1/16)]/(1/2)=15/16

Hence the answer is 11^15/16