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

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

Verify

Solution Steps

Convert 2 to fraction \frac{6}{3}.

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

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

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

Add 6 and 2 to get 8.

$$\sqrt{\frac{8}{3}}=2\sqrt{\frac{2}{3}}$$

Rewrite the square root of the division \sqrt{\frac{8}{3}} as the division of square roots \frac{\sqrt{8}}{\sqrt{3}}.

$$\frac{\sqrt{8}}{\sqrt{3}}=2\sqrt{\frac{2}{3}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

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

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

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

The square of \sqrt{3} is 3.

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

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

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

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

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

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

Subtract \frac{2\sqrt{6}}{3} from both sides.

$$\frac{2\sqrt{6}}{3}-\frac{2\sqrt{6}}{3}=0$$

Combine \frac{2\sqrt{6}}{3} and -\frac{2\sqrt{6}}{3} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$