Solve 5sqrt{1/5}*sqrt{20}-sqrt{40}divsqrt{8}

$$5 \sqrt{ \frac{ 1 }{ 5 } } \times \sqrt{ 20 } - \sqrt{ 40 } \div \sqrt{ 8 }$$

Evaluate

$$10-\sqrt{5}\approx 7.763932023$$

Solution Steps

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

$$5\times \frac{\sqrt{1}}{\sqrt{5}}\sqrt{20}-\frac{\sqrt{40}}{\sqrt{8}}$$

Calculate the square root of 1 and get 1.

$$5\times \frac{1}{\sqrt{5}}\sqrt{20}-\frac{\sqrt{40}}{\sqrt{8}}$$

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

$$5\times \frac{\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\sqrt{20}-\frac{\sqrt{40}}{\sqrt{8}}$$

The square of \sqrt{5} is 5.

$$5\times \frac{\sqrt{5}}{5}\sqrt{20}-\frac{\sqrt{40}}{\sqrt{8}}$$

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}.

$$5\times \frac{\sqrt{5}}{5}\times 2\sqrt{5}-\frac{\sqrt{40}}{\sqrt{8}}$$

Multiply 5 and 2 to get 10.

$$10\times \frac{\sqrt{5}}{5}\sqrt{5}-\frac{\sqrt{40}}{\sqrt{8}}$$

Cancel out 5, the greatest common factor in 10 and 5.

$$2\sqrt{5}\sqrt{5}-\frac{\sqrt{40}}{\sqrt{8}}$$

Rewrite the division of square roots \frac{\sqrt{40}}{\sqrt{8}} as the square root of the division \sqrt{\frac{40}{8}} and perform the division.

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

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

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

Multiply 2 and 5 to get 10.

$$10-\sqrt{5}$$