Solve 1/sqrt{2+1/sqrt{2-1/sqrt{2}}}

$$\frac { 1 } { \sqrt { 2 } + \frac { 1 } { \sqrt { 2 } - \frac { 1 } { \sqrt { 2 } } } }$$

Evaluate

$$\frac{\sqrt{2}}{4}\approx 0.353553391$$

Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{1}{\sqrt{2}+\frac{1}{\sqrt{2}-\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}}}$$

The square of \sqrt{2} is 2.

$$\frac{1}{\sqrt{2}+\frac{1}{\sqrt{2}-\frac{\sqrt{2}}{2}}}$$

Combine \sqrt{2} and -\frac{\sqrt{2}}{2} to get \frac{1}{2}\sqrt{2}.

$$\frac{1}{\sqrt{2}+\frac{1}{\frac{1}{2}\sqrt{2}}}$$

Rationalize the denominator of \frac{1}{\frac{1}{2}\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{1}{\sqrt{2}+\frac{\sqrt{2}}{\frac{1}{2}\left(\sqrt{2}\right)^{2}}}$$

The square of \sqrt{2} is 2.

$$\frac{1}{\sqrt{2}+\frac{\sqrt{2}}{\frac{1}{2}\times 2}}$$

Cancel out 2 and 2.

$$\frac{1}{\sqrt{2}+\frac{\sqrt{2}}{1}}$$

Anything divided by one gives itself.

$$\frac{1}{\sqrt{2}+\sqrt{2}}$$

Combine \sqrt{2} and \sqrt{2} to get 2\sqrt{2}.

$$\frac{1}{2\sqrt{2}}$$

Rationalize the denominator of \frac{1}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{\sqrt{2}}{4}$$