Solve sqrt{5}-3sqrt{20}+sqrt{125}+sqrt{1/5}

$$\sqrt { 5 } - 3 \sqrt { 20 } + \sqrt { 125 } + \sqrt { \frac { 1 } { 5 } }$$

Evaluate

$$\frac{\sqrt{5}}{5}\approx 0.447213595$$

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}.

$$\sqrt{5}-3\times 2\sqrt{5}+\sqrt{125}+\sqrt{\frac{1}{5}}$$

Multiply -3 and 2 to get -6.

$$\sqrt{5}-6\sqrt{5}+\sqrt{125}+\sqrt{\frac{1}{5}}$$

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

$$-5\sqrt{5}+\sqrt{125}+\sqrt{\frac{1}{5}}$$

Factor 125=5^{2}\times 5. Rewrite the square root of the product \sqrt{5^{2}\times 5} as the product of square roots \sqrt{5^{2}}\sqrt{5}. Take the square root of 5^{2}.

$$-5\sqrt{5}+5\sqrt{5}+\sqrt{\frac{1}{5}}$$

Combine -5\sqrt{5} and 5\sqrt{5} to get 0.

$$\sqrt{\frac{1}{5}}$$

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

$$\frac{\sqrt{1}}{\sqrt{5}}$$

Calculate the square root of 1 and get 1.

$$\frac{1}{\sqrt{5}}$$

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

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

The square of \sqrt{5} is 5.

$$\frac{\sqrt{5}}{5}$$