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

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

Evaluate

$$\sqrt{2}+\frac{3}{2}\approx 2.914213562$$

Solution Steps

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

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

The square of \sqrt{2} is 2.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2}{2}.

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

Since \frac{\sqrt{2}}{2} and \frac{2}{2} have the same denominator, add them by adding their numerators.

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

To raise \frac{\sqrt{2}+2}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{2}+2\right)^{2}.

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

The square of \sqrt{2} is 2.

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

Add 2 and 4 to get 6.

$$\frac{6+4\sqrt{2}}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{6+4\sqrt{2}}{4}$$

Expand

$$\sqrt{2} + \frac{3}{2} = 2.914213562$$

Solution Steps

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

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

The square of \sqrt{2} is 2.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2}{2}.

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

Since \frac{\sqrt{2}}{2} and \frac{2}{2} have the same denominator, add them by adding their numerators.

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

To raise \frac{\sqrt{2}+2}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{2}+2\right)^{2}.

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

The square of \sqrt{2} is 2.

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

Add 2 and 4 to get 6.

$$\frac{6+4\sqrt{2}}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{6+4\sqrt{2}}{4}$$