Solve 3/4sqrt{5/6}*-2/3sqrt{3}

$$\frac { 3 } { 4 } \sqrt { \frac { 5 } { 6 } } \cdot \frac { - 2 } { 3 } \sqrt { 3 }$$

Evaluate

$$-\frac{\sqrt{10}}{4}\approx -0.790569415$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{5}{6}} as the division of square roots \frac{\sqrt{5}}{\sqrt{6}}.

$$\frac{3}{4}\times \frac{\sqrt{5}}{\sqrt{6}}\times \frac{-2}{3}\sqrt{3}$$

Rationalize the denominator of \frac{\sqrt{5}}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{3}{4}\times \frac{\sqrt{5}\sqrt{6}}{\left(\sqrt{6}\right)^{2}}\times \frac{-2}{3}\sqrt{3}$$

The square of \sqrt{6} is 6.

$$\frac{3}{4}\times \frac{\sqrt{5}\sqrt{6}}{6}\times \frac{-2}{3}\sqrt{3}$$

To multiply \sqrt{5} and \sqrt{6}, multiply the numbers under the square root.

$$\frac{3}{4}\times \frac{\sqrt{30}}{6}\times \frac{-2}{3}\sqrt{3}$$

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

$$\frac{3}{4}\times \frac{\sqrt{30}}{6}\left(-\frac{2}{3}\right)\sqrt{3}$$

Multiply \frac{3}{4} times -\frac{2}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\left(-2\right)}{4\times 3}\times \frac{\sqrt{30}}{6}\sqrt{3}$$

Cancel out 3 in both numerator and denominator.

$$\frac{-2}{4}\times \frac{\sqrt{30}}{6}\sqrt{3}$$

Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.

$$-\frac{1}{2}\times \frac{\sqrt{30}}{6}\sqrt{3}$$

Multiply -\frac{1}{2} times \frac{\sqrt{30}}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-\sqrt{30}}{2\times 6}\sqrt{3}$$

Express \frac{-\sqrt{30}}{2\times 6}\sqrt{3} as a single fraction.

$$\frac{-\sqrt{30}\sqrt{3}}{2\times 6}$$

Factor 30=3\times 10. Rewrite the square root of the product \sqrt{3\times 10} as the product of square roots \sqrt{3}\sqrt{10}.

$$\frac{-\sqrt{3}\sqrt{10}\sqrt{3}}{2\times 6}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{-3\sqrt{10}}{2\times 6}$$

Multiply 2 and 6 to get 12.

$$\frac{-3\sqrt{10}}{12}$$

Divide -3\sqrt{10} by 12 to get -\frac{1}{4}\sqrt{10}.

$$-\frac{1}{4}\sqrt{10}$$