Solve 1/sqrt{2}+5/sqrt{5}

$$\frac { 1 } { \sqrt { 2 } } + \frac { 5 } { \sqrt { 5 } }$$

Evaluate

$$\frac{\sqrt{2}\left(\sqrt{10}+1\right)}{2}\approx 2.943174759$$

Short Solution Steps

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

$$\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+\frac{5}{\sqrt{5}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}}{2}+\frac{5}{\sqrt{5}}$$

Rationalize the denominator of \frac{5}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\sqrt{2}}{2}+\frac{5\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{2}}{2}+\frac{5\sqrt{5}}{5}$$

Cancel out 5 and 5.

$$\frac{\sqrt{2}}{2}+\sqrt{5}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{5} times \frac{2}{2}.

$$\frac{\sqrt{2}}{2}+\frac{2\sqrt{5}}{2}$$

Since \frac{\sqrt{2}}{2} and \frac{2\sqrt{5}}{2} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{2}+2\sqrt{5}}{2}$$