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

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

Evaluate

$$4-6\sqrt{5}\approx -9.416407865$$

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}.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

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

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

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

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

The square of \sqrt{6} is 6.

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

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

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

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

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

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

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

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

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

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

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

Cancel out -2 and -2.

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

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

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

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}.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -6\sqrt{5} times \frac{3}{3}.

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

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

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

Do the multiplications in 2\sqrt{6}\sqrt{6}+3\left(-6\right)\sqrt{5}.

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