Solve (5sqrt{1/6}-sqrt{20}-2sqrt{45})div(-sqrt{5})

$$( 5 \sqrt { \frac { 1 } { 6 } } - \sqrt { 20 } - 2 \sqrt { 45 } ) \div ( - \sqrt { 5 } )$$

Evaluate

$$-\frac{\sqrt{30}}{6}+8\approx 7.087129071$$

Short Solution Steps

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

$$\frac{5\times \frac{\sqrt{1}}{\sqrt{6}}-\sqrt{20}-2\sqrt{45}}{-\sqrt{5}}$$

Calculate the square root of 1 and get 1.

$$\frac{5\times \frac{1}{\sqrt{6}}-\sqrt{20}-2\sqrt{45}}{-\sqrt{5}}$$

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

$$\frac{5\times \frac{\sqrt{6}}{\left(\sqrt{6}\right)^{2}}-\sqrt{20}-2\sqrt{45}}{-\sqrt{5}}$$

The square of \sqrt{6} is 6.

$$\frac{5\times \frac{\sqrt{6}}{6}-\sqrt{20}-2\sqrt{45}}{-\sqrt{5}}$$

Express 5\times \frac{\sqrt{6}}{6} as a single fraction.

$$\frac{\frac{5\sqrt{6}}{6}-\sqrt{20}-2\sqrt{45}}{-\sqrt{5}}$$

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}.

$$\frac{\frac{5\sqrt{6}}{6}-2\sqrt{5}-2\sqrt{45}}{-\sqrt{5}}$$

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

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

Multiply -2 and 3 to get -6.

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

Combine -2\sqrt{5} and -6\sqrt{5} to get -8\sqrt{5}.

$$\frac{\frac{5\sqrt{6}}{6}-8\sqrt{5}}{-\sqrt{5}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -8\sqrt{5} times \frac{6}{6}.

$$\frac{\frac{5\sqrt{6}}{6}+\frac{6\left(-8\right)\sqrt{5}}{6}}{-\sqrt{5}}$$

Since \frac{5\sqrt{6}}{6} and \frac{6\left(-8\right)\sqrt{5}}{6} have the same denominator, add them by adding their numerators.

$$\frac{\frac{5\sqrt{6}+6\left(-8\right)\sqrt{5}}{6}}{-\sqrt{5}}$$

Do the multiplications in 5\sqrt{6}+6\left(-8\right)\sqrt{5}.

$$\frac{\frac{5\sqrt{6}-48\sqrt{5}}{6}}{-\sqrt{5}}$$

Express \frac{\frac{5\sqrt{6}-48\sqrt{5}}{6}}{-\sqrt{5}} as a single fraction.

$$\frac{5\sqrt{6}-48\sqrt{5}}{6\left(-\sqrt{5}\right)}$$

Multiply 6 and -1 to get -6.

$$\frac{5\sqrt{6}-48\sqrt{5}}{-6\sqrt{5}}$$

Rationalize the denominator of \frac{5\sqrt{6}-48\sqrt{5}}{-6\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\left(5\sqrt{6}-48\sqrt{5}\right)\sqrt{5}}{-6\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(5\sqrt{6}-48\sqrt{5}\right)\sqrt{5}}{-6\times 5}$$

Multiply -6 and 5 to get -30.

$$\frac{\left(5\sqrt{6}-48\sqrt{5}\right)\sqrt{5}}{-30}$$

Use the distributive property to multiply 5\sqrt{6}-48\sqrt{5} by \sqrt{5}.

$$\frac{5\sqrt{6}\sqrt{5}-48\left(\sqrt{5}\right)^{2}}{-30}$$

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

$$\frac{5\sqrt{30}-48\left(\sqrt{5}\right)^{2}}{-30}$$

The square of \sqrt{5} is 5.

$$\frac{5\sqrt{30}-48\times 5}{-30}$$

Multiply -48 and 5 to get -240.

$$\frac{5\sqrt{30}-240}{-30}$$