Solve 2/sqrt{5}+sqrt{20}+8/sqrt{80}

$$\frac { 2 } { \sqrt { 5 } } + \sqrt { 20 } + \frac { 8 } { \sqrt { 80 } }$$

Evaluate

$$\frac{14\sqrt{5}}{5}\approx 6.260990337$$

Solution Steps

Rationalize the denominator of \frac{2}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{2\sqrt{5}}{\left(\sqrt{5}\right)^{2}}+\sqrt{20}+\frac{8}{\sqrt{80}}$$

The square of \sqrt{5} is 5.

$$\frac{2\sqrt{5}}{5}+\sqrt{20}+\frac{8}{\sqrt{80}}$$

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}.

$$\frac{2\sqrt{5}}{5}+2\sqrt{5}+\frac{8}{\sqrt{80}}$$

Combine \frac{2\sqrt{5}}{5} and 2\sqrt{5} to get \frac{12}{5}\sqrt{5}.

$$\frac{12}{5}\sqrt{5}+\frac{8}{\sqrt{80}}$$

Factor 80=4^{2}\times 5. Rewrite the square root of the product \sqrt{4^{2}\times 5} as the product of square roots \sqrt{4^{2}}\sqrt{5}. Take the square root of 4^{2}.

$$\frac{12}{5}\sqrt{5}+\frac{8}{4\sqrt{5}}$$

Rationalize the denominator of \frac{8}{4\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{12}{5}\sqrt{5}+\frac{8\sqrt{5}}{4\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{12}{5}\sqrt{5}+\frac{8\sqrt{5}}{4\times 5}$$

Cancel out 4 in both numerator and denominator.

$$\frac{12}{5}\sqrt{5}+\frac{2\sqrt{5}}{5}$$

Combine \frac{12}{5}\sqrt{5} and \frac{2\sqrt{5}}{5} to get \frac{14}{5}\sqrt{5}.

$$\frac{14}{5}\sqrt{5}$$