Tons Of Square Roots

We have an equation: $\sqrt{\sqrt{2^5}\sqrt{2\sqrt{2\sqrt{2^6}}}} = ?$ , however, right now we will look only into a small chunk:

${2\sqrt{2^6}}$

It is easy to notice that $\sqrt{2^6}$ could be simplified to: ${2^3}$ , which would make: $2\times{2^3}$ and then: $2^4$ .

Now we have this: $\sqrt{\sqrt{2^5}\sqrt{2\sqrt{2^4}}} = ?$

Which simplifies to:

$\sqrt{\sqrt{2^5}\sqrt{2\times2^2}} = \sqrt{\sqrt{2^5}\sqrt{2^3}} = \sqrt{\sqrt{2^5\times2^3}} = \sqrt{\sqrt{2^8}} = \sqrt{2^4} = 2^2 = \boxed{4}$

$\sqrt { \sqrt { { 2 }^{ 5 } } \sqrt { 2\sqrt { 2\sqrt { { 2 }^{ 6 } } } } } =\sqrt { 4\sqrt { 2 } \sqrt { 2\sqrt { 2\times 8 } } } =\sqrt { 4\sqrt { 2 } \sqrt { 2\times 4 } } =\sqrt { 4\sqrt { 2 } \times \quad 2\sqrt { 2 } } =\sqrt { 4\times 2\times 2 } =\sqrt { 16 } =4$