Solve sqrt{2/3}(sqrt{6}-sqrt{3})

$$\sqrt { \frac { 2 } { 3 } } ( \sqrt { 6 } - \sqrt { 3 } )$$

Evaluate

$$2-\sqrt{2}\approx 0.585786438$$

Solution Steps

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

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

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

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

The square of \sqrt{3} is 3.

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

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

$$\frac{\sqrt{6}}{3}\left(\sqrt{6}-\sqrt{3}\right)$$

Express \frac{\sqrt{6}}{3}\left(\sqrt{6}-\sqrt{3}\right) as a single fraction.

$$\frac{\sqrt{6}\left(\sqrt{6}-\sqrt{3}\right)}{3}$$

Use the distributive property to multiply \sqrt{6} by \sqrt{6}-\sqrt{3}.

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

The square of \sqrt{6} is 6.

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

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}.

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

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

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

Divide each term of 6-3\sqrt{2} by 3 to get 2-\sqrt{2}.

$$2-\sqrt{2}$$