Simplify square root of 16^( square root of 2^( square root of 16))

$\sqrt{16^{\sqrt{2^{\sqrt{16}}}}}$

Step 1

Rewrite $16$ as $4^{2}$ .

$\sqrt{16^{\sqrt{2^{\sqrt{4^{2}}}}}}$

Step 2

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt{16^{\sqrt{2^{4}}}}$

Step 3

Raise $2$ to the power of $4$ .

$\sqrt{16^{\sqrt{16}}}$

Step 4

Rewrite $16$ as $4^{2}$ .

$\sqrt{16^{\sqrt{4^{2}}}}$

Step 5

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt{16^{4}}$

Step 6

Raise $16$ to the power of $4$ .

$\sqrt{65536}$

Step 7

Rewrite $65536$ as $256^{2}$ .

$\sqrt{256^{2}}$

Step 8

Pull terms out from under the radical, assuming positive real numbers.

$256$