Solve 4-6sqrt{frac{2/3}}{3sqrt{2/3}}

$$\frac{ 4-6 \sqrt{ \frac{ 2 }{ 3 } } }{ 3 \sqrt{ \frac{ 2 }{ 3 } } }$$

Evaluate

$$\frac{2\left(\sqrt{6}-3\right)}{3}\approx -0.367006838$$

Solution Steps

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

$$\frac{4-6\times \frac{\sqrt{2}}{\sqrt{3}}}{3\sqrt{\frac{2}{3}}}$$

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

$$\frac{4-6\times \frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}{3\sqrt{\frac{2}{3}}}$$

The square of \sqrt{3} is 3.

$$\frac{4-6\times \frac{\sqrt{2}\sqrt{3}}{3}}{3\sqrt{\frac{2}{3}}}$$

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

$$\frac{4-6\times \frac{\sqrt{6}}{3}}{3\sqrt{\frac{2}{3}}}$$

Cancel out 3, the greatest common factor in 6 and 3.

$$\frac{4-2\sqrt{6}}{3\sqrt{\frac{2}{3}}}$$

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

$$\frac{4-2\sqrt{6}}{3\times \frac{\sqrt{2}}{\sqrt{3}}}$$

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

$$\frac{4-2\sqrt{6}}{3\times \frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}$$

The square of \sqrt{3} is 3.

$$\frac{4-2\sqrt{6}}{3\times \frac{\sqrt{2}\sqrt{3}}{3}}$$

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

$$\frac{4-2\sqrt{6}}{3\times \frac{\sqrt{6}}{3}}$$

Cancel out 3 and 3.

$$\frac{4-2\sqrt{6}}{\sqrt{6}}$$

Rationalize the denominator of \frac{4-2\sqrt{6}}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\left(4-2\sqrt{6}\right)\sqrt{6}}{\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\left(4-2\sqrt{6}\right)\sqrt{6}}{6}$$

Use the distributive property to multiply 4-2\sqrt{6} by \sqrt{6}.

$$\frac{4\sqrt{6}-2\left(\sqrt{6}\right)^{2}}{6}$$

The square of \sqrt{6} is 6.

$$\frac{4\sqrt{6}-2\times 6}{6}$$

Multiply -2 and 6 to get -12.

$$\frac{4\sqrt{6}-12}{6}$$

Factor

$$\frac{2 {(\sqrt{6} - 3)}}{3} = -0.3670068381445481$$