Solve 1/3sqrt{2}(sqrt{2}-1)+1/3(sqrt{3}+sqrt{2})-2/3

$$\frac { 1 } { 3 } \sqrt { 2 } ( \sqrt { 2 } - 1 ) + \frac { 1 } { 3 } ( \sqrt { 3 } + \sqrt { 2 } ) - \frac { 2 } { 3 }$$

Evaluate

$$\frac{\sqrt{3}}{3}\approx 0.577350269$$

Solution Steps

Use the distributive property to multiply \frac{1}{3}\sqrt{2} by \sqrt{2}-1.

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{1}{3}\times 2+\frac{1}{3}\sqrt{2}\left(-1\right)+\frac{1}{3}\left(\sqrt{3}+\sqrt{2}\right)-\frac{2}{3}$$

Multiply \frac{1}{3} and 2 to get \frac{2}{3}.

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

Multiply \frac{1}{3} and -1 to get -\frac{1}{3}.

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

Use the distributive property to multiply \frac{1}{3} by \sqrt{3}+\sqrt{2}.

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

Combine -\frac{1}{3}\sqrt{2} and \frac{1}{3}\sqrt{2} to get 0.

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

Subtract \frac{2}{3} from \frac{2}{3} to get 0.

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