Solve sqrt{6/9+1/3+6/9}

$$\sqrt { \frac { 6 } { 9 } + \frac { 1 } { 3 } + \frac { 6 } { 9 } }$$

Evaluate

$$\frac{\sqrt{15}}{3}\approx 1.290994449$$

Solution Steps

Reduce the fraction \frac{6}{9} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{2}{3}+\frac{1}{3}+\frac{6}{9}}$$

Since \frac{2}{3} and \frac{1}{3} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{2+1}{3}+\frac{6}{9}}$$

Add 2 and 1 to get 3.

$$\sqrt{\frac{3}{3}+\frac{6}{9}}$$

Divide 3 by 3 to get 1.

$$\sqrt{1+\frac{6}{9}}$$

Reduce the fraction \frac{6}{9} to lowest terms by extracting and canceling out 3.

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

Convert 1 to fraction \frac{3}{3}.

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

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

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

Add 3 and 2 to get 5.

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

Rewrite the square root of the division \sqrt{\frac{5}{3}} as the division of square roots \frac{\sqrt{5}}{\sqrt{3}}.

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

Rationalize the denominator of \frac{\sqrt{5}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\sqrt{15}}{3}$$