Solve 2sqrt{2/3}frac{sqrt{3}}{3}

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

Evaluate

$$\frac{2\sqrt{2}}{3}\approx 0.942809042$$

Solution Steps

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

Express 2\times \frac{\sqrt{6}}{3} as a single fraction.

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

Multiply \frac{2\sqrt{6}}{3} times \frac{\sqrt{3}}{3} by multiplying numerator times numerator and denominator times denominator.

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

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

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

Multiply \sqrt{3} and \sqrt{3} to get 3.

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

Multiply 2 and 3 to get 6.

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

Multiply 3 and 3 to get 9.

$$\frac{6\sqrt{2}}{9}$$

Divide 6\sqrt{2} by 9 to get \frac{2}{3}\sqrt{2}.

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