Solve sqrt{2/3}+sqrt{1/27}

$$\sqrt { \frac { 2 } { 3 } } + \sqrt { \frac { 1 } { 27 } }$$

Evaluate

$$\frac{\sqrt{3}\left(3\sqrt{2}+1\right)}{9}\approx 1.008946671$$

Short Solution Steps

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

$$\frac{\sqrt{2}}{\sqrt{3}}+\sqrt{\frac{1}{27}}$$

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

$$\frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\sqrt{\frac{1}{27}}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{2}\sqrt{3}}{3}+\sqrt{\frac{1}{27}}$$

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

$$\frac{\sqrt{6}}{3}+\sqrt{\frac{1}{27}}$$

Rewrite the square root of the division \sqrt{\frac{1}{27}} as the division of square roots \frac{\sqrt{1}}{\sqrt{27}}.

$$\frac{\sqrt{6}}{3}+\frac{\sqrt{1}}{\sqrt{27}}$$

Calculate the square root of 1 and get 1.

$$\frac{\sqrt{6}}{3}+\frac{1}{\sqrt{27}}$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

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

Rationalize the denominator of \frac{1}{3\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

Multiply 3 and 3 to get 9.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 9 is 9. Multiply \frac{\sqrt{6}}{3} times \frac{3}{3}.

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

Since \frac{3\sqrt{6}}{9} and \frac{\sqrt{3}}{9} have the same denominator, add them by adding their numerators.

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