Solve sqrt{frac{2/3}}{-sqrt{7/3}}

$$\frac{ \sqrt{ \frac{ 2 }{ 3 } } }{ - \sqrt{ \frac{ 7 }{ 3 } } }$$

Evaluate

$$-\frac{\sqrt{14}}{7}\approx -0.534522484$$

Solution Steps

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

$$\frac{\frac{\sqrt{2}}{\sqrt{3}}}{-\sqrt{\frac{7}{3}}}$$

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

$$\frac{\frac{\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}{-\sqrt{\frac{7}{3}}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{\sqrt{2}\sqrt{3}}{3}}{-\sqrt{\frac{7}{3}}}$$

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

$$\frac{\frac{\sqrt{6}}{3}}{-\sqrt{\frac{7}{3}}}$$

Rewrite the square root of the division \sqrt{\frac{7}{3}} as the division of square roots \frac{\sqrt{7}}{\sqrt{3}}.

$$\frac{\frac{\sqrt{6}}{3}}{-\frac{\sqrt{7}}{\sqrt{3}}}$$

Rationalize the denominator of \frac{\sqrt{7}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\frac{\sqrt{6}}{3}}{-\frac{\sqrt{7}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{\sqrt{6}}{3}}{-\frac{\sqrt{7}\sqrt{3}}{3}}$$

To multiply \sqrt{7} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\frac{\sqrt{6}}{3}}{-\frac{\sqrt{21}}{3}}$$

Express \frac{\frac{\sqrt{6}}{3}}{-\frac{\sqrt{21}}{3}} as a single fraction.

$$\frac{\sqrt{6}}{3\left(-\frac{\sqrt{21}}{3}\right)}$$

Cancel out 3 and 3.

$$\frac{\sqrt{6}}{-\sqrt{21}}$$

Rationalize the denominator of \frac{\sqrt{6}}{-\sqrt{21}} by multiplying numerator and denominator by \sqrt{21}.

$$\frac{\sqrt{6}\sqrt{21}}{-\left(\sqrt{21}\right)^{2}}$$

The square of \sqrt{21} is 21.

$$\frac{\sqrt{6}\sqrt{21}}{-21}$$

To multiply \sqrt{6} and \sqrt{21}, multiply the numbers under the square root.

$$\frac{\sqrt{126}}{-21}$$

Factor 126=3^{2}\times 14. Rewrite the square root of the product \sqrt{3^{2}\times 14} as the product of square roots \sqrt{3^{2}}\sqrt{14}. Take the square root of 3^{2}.

$$\frac{3\sqrt{14}}{-21}$$

Divide 3\sqrt{14} by -21 to get -\frac{1}{7}\sqrt{14}.

$$-\frac{1}{7}\sqrt{14}$$