Solve frac{sqrt{5}+sqrt{90}}{sqrt{20}}

$$\frac { \sqrt { 5 } + \sqrt { 90 } } { \sqrt { 20 } }$$

Evaluate

$$\frac{3\sqrt{2}+1}{2}\approx 2.621320344$$

Solution Steps

Factor 90=3^{2}\times 10. Rewrite the square root of the product \sqrt{3^{2}\times 10} as the product of square roots \sqrt{3^{2}}\sqrt{10}. Take the square root of 3^{2}.

$$\frac{\sqrt{5}+3\sqrt{10}}{\sqrt{20}}$$

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}.

$$\frac{\sqrt{5}+3\sqrt{10}}{2\sqrt{5}}$$

Rationalize the denominator of \frac{\sqrt{5}+3\sqrt{10}}{2\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\left(\sqrt{5}+3\sqrt{10}\right)\sqrt{5}}{2\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(\sqrt{5}+3\sqrt{10}\right)\sqrt{5}}{2\times 5}$$

Multiply 2 and 5 to get 10.

$$\frac{\left(\sqrt{5}+3\sqrt{10}\right)\sqrt{5}}{10}$$

Use the distributive property to multiply \sqrt{5}+3\sqrt{10} by \sqrt{5}.

$$\frac{\left(\sqrt{5}\right)^{2}+3\sqrt{10}\sqrt{5}}{10}$$

The square of \sqrt{5} is 5.

$$\frac{5+3\sqrt{10}\sqrt{5}}{10}$$

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}.

$$\frac{5+3\sqrt{5}\sqrt{2}\sqrt{5}}{10}$$

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

$$\frac{5+3\times 5\sqrt{2}}{10}$$

Multiply 3 and 5 to get 15.

$$\frac{5+15\sqrt{2}}{10}$$