Solve {left(sqrt{2}+1/sqrt{2}right)}^2+1/2

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

Evaluate

$$5$$

Solution Steps

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

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

The square of \sqrt{2} is 2.

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

Combine \sqrt{2} and \frac{\sqrt{2}}{2} to get \frac{3}{2}\sqrt{2}.

$$\left(\frac{3}{2}\sqrt{2}\right)^{2}+\frac{1}{2}$$

Expand \left(\frac{3}{2}\sqrt{2}\right)^{2}.

$$\left(\frac{3}{2}\right)^{2}\left(\sqrt{2}\right)^{2}+\frac{1}{2}$$

Calculate \frac{3}{2} to the power of 2 and get \frac{9}{4}.

$$\frac{9}{4}\left(\sqrt{2}\right)^{2}+\frac{1}{2}$$

The square of \sqrt{2} is 2.

$$\frac{9}{4}\times 2+\frac{1}{2}$$

Multiply \frac{9}{4} and 2 to get \frac{9}{2}.

$$\frac{9}{2}+\frac{1}{2}$$

Add \frac{9}{2} and \frac{1}{2} to get 5.

$$5$$

Factor

$$5$$