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

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

Evaluate

$$-\frac{2\sqrt{6}}{3}-2\approx -3.632993162$$

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

$$\frac{2 {(-\sqrt{6} - 3)}}{3} = -3.632993161855452$$