Solve sqrt{24}(-sqrt{2/3}+3sqrt{5/6}+sqrt{5})

$$\sqrt { 24 } ( - \sqrt { \frac { 2 } { 3 } } + 3 \sqrt { \frac { 5 } { 6 } } + \sqrt { 5 } )$$

Evaluate

$$2\sqrt{30}+6\sqrt{5}-4\approx 20.370859015$$

Short Solution Steps

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

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

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

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

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

The square of \sqrt{6} is 6.

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

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

$$2\sqrt{6}\left(-\frac{\sqrt{6}}{3}+3\times \frac{\sqrt{30}}{6}+\sqrt{5}\right)$$

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

$$2\sqrt{6}\left(-\frac{\sqrt{6}}{3}+\frac{\sqrt{30}}{2}+\sqrt{5}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply -\frac{\sqrt{6}}{3} times \frac{2}{2}. Multiply \frac{\sqrt{30}}{2} times \frac{3}{3}.

$$2\sqrt{6}\left(-\frac{2\sqrt{6}}{6}+\frac{3\sqrt{30}}{6}+\sqrt{5}\right)$$

Since -\frac{2\sqrt{6}}{6} and \frac{3\sqrt{30}}{6} have the same denominator, add them by adding their numerators.

$$2\sqrt{6}\left(\frac{-2\sqrt{6}+3\sqrt{30}}{6}+\sqrt{5}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{5} times \frac{6}{6}.

$$2\sqrt{6}\left(\frac{-2\sqrt{6}+3\sqrt{30}}{6}+\frac{6\sqrt{5}}{6}\right)$$

Since \frac{-2\sqrt{6}+3\sqrt{30}}{6} and \frac{6\sqrt{5}}{6} have the same denominator, add them by adding their numerators.

$$2\sqrt{6}\times \frac{-2\sqrt{6}+3\sqrt{30}+6\sqrt{5}}{6}$$

Cancel out 6, the greatest common factor in 2 and 6.

$$\frac{-2\sqrt{6}+3\sqrt{30}+6\sqrt{5}}{3}\sqrt{6}$$

Express \frac{-2\sqrt{6}+3\sqrt{30}+6\sqrt{5}}{3}\sqrt{6} as a single fraction.

$$\frac{\left(-2\sqrt{6}+3\sqrt{30}+6\sqrt{5}\right)\sqrt{6}}{3}$$

Use the distributive property to multiply -2\sqrt{6}+3\sqrt{30}+6\sqrt{5} by \sqrt{6}.

$$\frac{-2\left(\sqrt{6}\right)^{2}+3\sqrt{30}\sqrt{6}+6\sqrt{5}\sqrt{6}}{3}$$

The square of \sqrt{6} is 6.

$$\frac{-2\times 6+3\sqrt{30}\sqrt{6}+6\sqrt{5}\sqrt{6}}{3}$$

Multiply -2 and 6 to get -12.

$$\frac{-12+3\sqrt{30}\sqrt{6}+6\sqrt{5}\sqrt{6}}{3}$$

Factor 30=6\times 5. Rewrite the square root of the product \sqrt{6\times 5} as the product of square roots \sqrt{6}\sqrt{5}.

$$\frac{-12+3\sqrt{6}\sqrt{5}\sqrt{6}+6\sqrt{5}\sqrt{6}}{3}$$

Multiply \sqrt{6} and \sqrt{6} to get 6.

$$\frac{-12+3\times 6\sqrt{5}+6\sqrt{5}\sqrt{6}}{3}$$

Multiply 3 and 6 to get 18.

$$\frac{-12+18\sqrt{5}+6\sqrt{5}\sqrt{6}}{3}$$

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

$$\frac{-12+18\sqrt{5}+6\sqrt{30}}{3}$$