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

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

Evaluate

$$\frac{\sqrt{6}}{6}\approx 0.40824829$$

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}}.

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

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

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

The square of \sqrt{3} is 3.

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

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

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

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

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

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

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

The square of \sqrt{2} is 2.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{6} times \frac{3}{3}.

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

Since \frac{3\sqrt{6}}{3} and \frac{\sqrt{6}}{3} have the same denominator, subtract them by subtracting their numerators.

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

Do the calculations in 3\sqrt{6}-\sqrt{6}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{2\sqrt{6}}{3} times \frac{2}{2}. Multiply \frac{\sqrt{6}}{2} times \frac{3}{3}.

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

Since \frac{2\times 2\sqrt{6}}{6} and \frac{3\sqrt{6}}{6} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2\times 2\sqrt{6}-3\sqrt{6}.

$$\frac{4\sqrt{6}-3\sqrt{6}}{6}$$

Do the calculations in 4\sqrt{6}-3\sqrt{6}.

$$\frac{\sqrt{6}}{6}$$