Solve frac{sqrt{10}+sqrt{2}+sqrt{3}}{sqrt{5}}

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

Evaluate

$$\frac{\sqrt{5}\left(\sqrt{2}+\sqrt{3}+\sqrt{10}\right)}{5}\approx 2.821265764$$

Solution Steps

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

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

The square of \sqrt{5} is 5.

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

Use the distributive property to multiply \sqrt{10}+\sqrt{2}+\sqrt{3} by \sqrt{5}.

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

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}.

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

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

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

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

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

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

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