Solve sqrt{1/2+frac{1/3}}{sqrt{10/3}}

$$\frac { \sqrt { \frac { 1 } { 2 } + \frac { 1 } { 3 } } } { \sqrt { \frac { 10 } { 3 } } }$$

Evaluate

$$\frac{1}{2}=0.5$$

Solution Steps

Least common multiple of 2 and 3 is 6. Convert \frac{1}{2} and \frac{1}{3} to fractions with denominator 6.

$$\frac{\sqrt{\frac{3}{6}+\frac{2}{6}}}{\sqrt{\frac{10}{3}}}$$

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

$$\frac{\sqrt{\frac{3+2}{6}}}{\sqrt{\frac{10}{3}}}$$

Add 3 and 2 to get 5.

$$\frac{\sqrt{\frac{5}{6}}}{\sqrt{\frac{10}{3}}}$$

Rewrite the square root of the division \sqrt{\frac{5}{6}} as the division of square roots \frac{\sqrt{5}}{\sqrt{6}}.

$$\frac{\frac{\sqrt{5}}{\sqrt{6}}}{\sqrt{\frac{10}{3}}}$$

Rationalize the denominator of \frac{\sqrt{5}}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\frac{\sqrt{5}\sqrt{6}}{\left(\sqrt{6}\right)^{2}}}{\sqrt{\frac{10}{3}}}$$

The square of \sqrt{6} is 6.

$$\frac{\frac{\sqrt{5}\sqrt{6}}{6}}{\sqrt{\frac{10}{3}}}$$

To multiply \sqrt{5} and \sqrt{6}, multiply the numbers under the square root.

$$\frac{\frac{\sqrt{30}}{6}}{\sqrt{\frac{10}{3}}}$$

Rewrite the square root of the division \sqrt{\frac{10}{3}} as the division of square roots \frac{\sqrt{10}}{\sqrt{3}}.

$$\frac{\frac{\sqrt{30}}{6}}{\frac{\sqrt{10}}{\sqrt{3}}}$$

Rationalize the denominator of \frac{\sqrt{10}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\frac{\sqrt{30}}{6}}{\frac{\sqrt{10}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{\sqrt{30}}{6}}{\frac{\sqrt{10}\sqrt{3}}{3}}$$

To multiply \sqrt{10} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\frac{\sqrt{30}}{6}}{\frac{\sqrt{30}}{3}}$$

Divide \frac{\sqrt{30}}{6} by \frac{\sqrt{30}}{3} by multiplying \frac{\sqrt{30}}{6} by the reciprocal of \frac{\sqrt{30}}{3}.

$$\frac{\sqrt{30}\times 3}{6\sqrt{30}}$$

Cancel out 3\sqrt{30} in both numerator and denominator.

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

Factor

$$\frac{1}{2} = 0.5$$