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

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

Evaluate

$$7$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{5}} as the division of square roots \frac{\sqrt{1}}{\sqrt{5}}.

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

Calculate the square root of 1 and get 1.

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

Rationalize the denominator of \frac{1}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

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

The square of \sqrt{5} is 5.

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

Cancel out 5 and 5.

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

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}.

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

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

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

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}.

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

Multiply -2 and 3 to get -6.

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

Combine -\sqrt{5} and -6\sqrt{5} to get -7\sqrt{5}.

$$\frac{-7\sqrt{5}}{-\sqrt{5}}$$

Cancel out \sqrt{5} in both numerator and denominator.

$$\frac{-7}{-1}$$

Fraction \frac{-7}{-1} can be simplified to 7 by removing the negative sign from both the numerator and the denominator.

$$7$$

Factor

$$7$$