Solve sqrt{13/5}*2sqrt{3}*(-3/4sqrt{1/6})

$$\sqrt { 1 \frac { 3 } { 5 } } \times 2 \sqrt { 3 } \times ( - \frac { 3 } { 4 } \sqrt { \frac { 1 } { 6 } } )$$

Evaluate

$$-\frac{3\sqrt{5}}{5}\approx -1.341640786$$

Solution Steps

Multiply 1 and 5 to get 5.

$$2\sqrt{\frac{5+3}{5}}\sqrt{3}\left(-\frac{3}{4}\right)\sqrt{\frac{1}{6}}$$

Add 5 and 3 to get 8.

$$2\sqrt{\frac{8}{5}}\sqrt{3}\left(-\frac{3}{4}\right)\sqrt{\frac{1}{6}}$$

Rewrite the square root of the division \sqrt{\frac{8}{5}} as the division of square roots \frac{\sqrt{8}}{\sqrt{5}}.

$$2\times \frac{\sqrt{8}}{\sqrt{5}}\sqrt{3}\left(-\frac{3}{4}\right)\sqrt{\frac{1}{6}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$2\times \frac{2\sqrt{2}}{\sqrt{5}}\sqrt{3}\left(-\frac{3}{4}\right)\sqrt{\frac{1}{6}}$$

Rationalize the denominator of \frac{2\sqrt{2}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$2\times \frac{2\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\sqrt{3}\left(-\frac{3}{4}\right)\sqrt{\frac{1}{6}}$$

The square of \sqrt{5} is 5.

$$2\times \frac{2\sqrt{2}\sqrt{5}}{5}\sqrt{3}\left(-\frac{3}{4}\right)\sqrt{\frac{1}{6}}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$2\times \frac{2\sqrt{10}}{5}\sqrt{3}\left(-\frac{3}{4}\right)\sqrt{\frac{1}{6}}$$

Express 2\left(-\frac{3}{4}\right) as a single fraction.

$$\frac{2\left(-3\right)}{4}\times \frac{2\sqrt{10}}{5}\sqrt{3}\sqrt{\frac{1}{6}}$$

Multiply 2 and -3 to get -6.

$$\frac{-6}{4}\times \frac{2\sqrt{10}}{5}\sqrt{3}\sqrt{\frac{1}{6}}$$

Reduce the fraction \frac{-6}{4} to lowest terms by extracting and canceling out 2.

$$-\frac{3}{2}\times \frac{2\sqrt{10}}{5}\sqrt{3}\sqrt{\frac{1}{6}}$$

Rewrite the square root of the division \sqrt{\frac{1}{6}} as the division of square roots \frac{\sqrt{1}}{\sqrt{6}}.

$$-\frac{3}{2}\times \frac{2\sqrt{10}}{5}\sqrt{3}\times \frac{\sqrt{1}}{\sqrt{6}}$$

Calculate the square root of 1 and get 1.

$$-\frac{3}{2}\times \frac{2\sqrt{10}}{5}\sqrt{3}\times \frac{1}{\sqrt{6}}$$

Rationalize the denominator of \frac{1}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$-\frac{3}{2}\times \frac{2\sqrt{10}}{5}\sqrt{3}\times \frac{\sqrt{6}}{\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$-\frac{3}{2}\times \frac{2\sqrt{10}}{5}\sqrt{3}\times \frac{\sqrt{6}}{6}$$

Multiply -\frac{3}{2} times \frac{2\sqrt{10}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-3\times 2\sqrt{10}}{2\times 5}\sqrt{3}\times \frac{\sqrt{6}}{6}$$

Cancel out 2 in both numerator and denominator.

$$\frac{-3\sqrt{10}}{5}\sqrt{3}\times \frac{\sqrt{6}}{6}$$

Express \frac{-3\sqrt{10}}{5}\sqrt{3} as a single fraction.

$$\frac{-3\sqrt{10}\sqrt{3}}{5}\times \frac{\sqrt{6}}{6}$$

Multiply \frac{-3\sqrt{10}\sqrt{3}}{5} times \frac{\sqrt{6}}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-3\sqrt{10}\sqrt{3}\sqrt{6}}{5\times 6}$$

Cancel out 3 in both numerator and denominator.

$$\frac{-\sqrt{3}\sqrt{6}\sqrt{10}}{2\times 5}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$\frac{-\sqrt{3}\sqrt{3}\sqrt{2}\sqrt{10}}{2\times 5}$$

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

$$\frac{-3\sqrt{2}\sqrt{10}}{2\times 5}$$

Factor 10=2\times 5. Rewrite the square root of the product \sqrt{2\times 5} as the product of square roots \sqrt{2}\sqrt{5}.

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

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

Multiply -3 and 2 to get -6.

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

Multiply 2 and 5 to get 10.

$$\frac{-6\sqrt{5}}{10}$$

Divide -6\sqrt{5} by 10 to get -\frac{3}{5}\sqrt{5}.

$$-\frac{3}{5}\sqrt{5}$$