Solve frac{sqrt{5}}{sqrt{20}+sqrt{5}}

$$\frac { \sqrt { 5 } } { \sqrt { 20 } + \sqrt { 5 } }$$

Evaluate

$$\frac{1}{3}\approx 0.333333333$$

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}.

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

Combine 2\sqrt{5} and \sqrt{5} to get 3\sqrt{5}.

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

Rationalize the denominator of \frac{\sqrt{5}}{3\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

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

The square of \sqrt{5} is 5.

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

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

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

Multiply 3 and 5 to get 15.

$$\frac{5}{15}$$

Reduce the fraction \frac{5}{15} to lowest terms by extracting and canceling out 5.

$$\frac{1}{3}$$

Factor

$$\frac{1}{3} = 0.3333333333333333$$