Solve sqrt{15}*frac{sqrt{5}}{3}*sqrt{20}div(-1/3sqrt{6})

$$\sqrt { 15 } \times \frac { \sqrt { 5 } } { 3 } \times \sqrt { 20 } \div ( - \frac { 1 } { 3 } \sqrt { 6 } )$$

Evaluate

$$-5\sqrt{10}\approx -15.811388301$$

Solution Steps

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\frac{\sqrt{15}\times \frac{\sqrt{5}}{3}\times 2\sqrt{5}}{-\frac{1}{3}\sqrt{6}}$$

Express \sqrt{15}\times \frac{\sqrt{5}}{3} as a single fraction.

$$\frac{\frac{\sqrt{15}\sqrt{5}}{3}\times 2\sqrt{5}}{-\frac{1}{3}\sqrt{6}}$$

Express \frac{\sqrt{15}\sqrt{5}}{3}\times 2 as a single fraction.

$$\frac{\frac{\sqrt{15}\sqrt{5}\times 2}{3}\sqrt{5}}{-\frac{1}{3}\sqrt{6}}$$

Express \frac{\sqrt{15}\sqrt{5}\times 2}{3}\sqrt{5} as a single fraction.

$$\frac{\frac{\sqrt{15}\sqrt{5}\times 2\sqrt{5}}{3}}{-\frac{1}{3}\sqrt{6}}$$

Express \frac{\frac{\sqrt{15}\sqrt{5}\times 2\sqrt{5}}{3}}{-\frac{1}{3}\sqrt{6}} as a single fraction.

$$\frac{\sqrt{15}\sqrt{5}\times 2\sqrt{5}}{3\left(-\frac{1}{3}\right)\sqrt{6}}$$

Rationalize the denominator of \frac{\sqrt{15}\sqrt{5}\times 2\sqrt{5}}{3\left(-\frac{1}{3}\right)\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\sqrt{15}\sqrt{5}\times 2\sqrt{5}\sqrt{6}}{3\left(-\frac{1}{3}\right)\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{\sqrt{15}\sqrt{5}\times 2\sqrt{5}\sqrt{6}}{3\left(-\frac{1}{3}\right)\times 6}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{\sqrt{15}\times 5\times 2\sqrt{6}}{3\left(-\frac{1}{3}\right)\times 6}$$

Multiply 5 and 2 to get 10.

$$\frac{\sqrt{15}\times 10\sqrt{6}}{3\left(-\frac{1}{3}\right)\times 6}$$

To multiply \sqrt{15} and \sqrt{6}, multiply the numbers under the square root.

$$\frac{\sqrt{90}\times 10}{3\left(-\frac{1}{3}\right)\times 6}$$

Cancel out 3 and 3.

$$\frac{\sqrt{90}\times 10}{-6}$$

Factor 90=3^{2}\times 10. Rewrite the square root of the product \sqrt{3^{2}\times 10} as the product of square roots \sqrt{3^{2}}\sqrt{10}. Take the square root of 3^{2}.

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

Multiply 3 and 10 to get 30.

$$\frac{30\sqrt{10}}{-6}$$

Divide 30\sqrt{10} by -6 to get -5\sqrt{10}.

$$-5\sqrt{10}$$