Solve sqrt{1/2}+sqrt{2/3}

$$\sqrt { \frac { 1 } { 2 } } + \sqrt { \frac { 2 } { 3 } }$$

Evaluate

$$\frac{\sqrt{2}\left(2\sqrt{3}+3\right)}{6}\approx 1.523603362$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

$$\frac{\sqrt{1}}{\sqrt{2}}+\sqrt{\frac{2}{3}}$$

Calculate the square root of 1 and get 1.

$$\frac{1}{\sqrt{2}}+\sqrt{\frac{2}{3}}$$

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

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

The square of \sqrt{2} is 2.

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

Rewrite the square root of the division \sqrt{\frac{2}{3}} as the division of square roots \frac{\sqrt{2}}{\sqrt{3}}.

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

Rationalize the denominator of \frac{\sqrt{2}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\sqrt{2}}{2}+\frac{\sqrt{6}}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 3 is 6. Multiply \frac{\sqrt{2}}{2} times \frac{3}{3}. Multiply \frac{\sqrt{6}}{3} times \frac{2}{2}.

$$\frac{3\sqrt{2}}{6}+\frac{2\sqrt{6}}{6}$$

Since \frac{3\sqrt{2}}{6} and \frac{2\sqrt{6}}{6} have the same denominator, add them by adding their numerators.

$$\frac{3\sqrt{2}+2\sqrt{6}}{6}$$