Solve frac{sqrt{2}+sqrt{5}}{sqrt{10}}

$$\frac { \sqrt { 2 } + \sqrt { 5 } } { \sqrt { 10 } }$$

Evaluate

$$\frac{2\sqrt{5}+5\sqrt{2}}{10}\approx 1.154320377$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{2}+\sqrt{5}}{\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{\left(\sqrt{2}+\sqrt{5}\right)\sqrt{10}}{\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{\left(\sqrt{2}+\sqrt{5}\right)\sqrt{10}}{10}$$

Use the distributive property to multiply \sqrt{2}+\sqrt{5} by \sqrt{10}.

$$\frac{\sqrt{2}\sqrt{10}+\sqrt{5}\sqrt{10}}{10}$$

Factor 10=2\times 5. Rewrite the square root of the product \sqrt{2\times 5} as the product of square roots \sqrt{2}\sqrt{5}.

$$\frac{\sqrt{2}\sqrt{2}\sqrt{5}+\sqrt{5}\sqrt{10}}{10}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{2\sqrt{5}+\sqrt{5}\sqrt{10}}{10}$$

Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.

$$\frac{2\sqrt{5}+\sqrt{5}\sqrt{5}\sqrt{2}}{10}$$

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

$$\frac{2\sqrt{5}+5\sqrt{2}}{10}$$