Solve {r}{2sqrt{20}-sqrt{5}}{+3sqrt{1/5}}

$$\left. \begin{array} { r } { 2 \sqrt { 20 } - \sqrt { 5 } } \\ { + 3 \sqrt { \frac { 1 } { 5 } } } \end{array} \right.$$

Evaluate

$$\frac{18\sqrt{5}}{5}\approx 8.049844719$$

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}.

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

Multiply 2 and 2 to get 4.

$$4\sqrt{5}-\sqrt{5}+3\sqrt{\frac{1}{5}}$$

Combine 4\sqrt{5} and -\sqrt{5} to get 3\sqrt{5}.

$$3\sqrt{5}+3\sqrt{\frac{1}{5}}$$

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

$$3\sqrt{5}+3\times \frac{\sqrt{1}}{\sqrt{5}}$$

Calculate the square root of 1 and get 1.

$$3\sqrt{5}+3\times \frac{1}{\sqrt{5}}$$

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

$$3\sqrt{5}+3\times \frac{\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$3\sqrt{5}+3\times \frac{\sqrt{5}}{5}$$

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

$$3\sqrt{5}+\frac{3\sqrt{5}}{5}$$

Combine 3\sqrt{5} and \frac{3\sqrt{5}}{5} to get \frac{18}{5}\sqrt{5}.

$$\frac{18}{5}\sqrt{5}$$