Solve sqrt{sqrt{sqrt{sqrt{sqrt{sqrt{left(2^2right)^left(2^2right)^2}}}}}}

$$\sqrt{ \sqrt{ \sqrt{ \sqrt{ \sqrt{ \sqrt{ { \left( { 2 }^{ 2 } \right) }^{ { \left( { 2 }^{ 2 } \right) }^{ 2 } } } } } } } }$$

Evaluate

$$\sqrt{2}\approx 1.414213562$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\left(2^{2}\right)^{2^{4}}}}}}}}$$

Calculate 2 to the power of 2 and get 4.

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

Calculate 2 to the power of 4 and get 16.

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

Calculate 4 to the power of 16 and get 4294967296.

$$\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{4294967296}}}}}}$$

Calculate the square root of 4294967296 and get 65536.

$$\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{65536}}}}}$$

Calculate the square root of 65536 and get 256.

$$\sqrt{\sqrt{\sqrt{\sqrt{256}}}}$$

Calculate the square root of 256 and get 16.

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

Calculate the square root of 16 and get 4.

$$\sqrt{\sqrt{4}}$$

Calculate the square root of 4 and get 2.

$$\sqrt{2}$$