Solve frac{sqrt{7}+sqrt{8}}{sqrt{10}}

$$\frac { \sqrt { 7 } + \sqrt { 8 } } { \sqrt { 10 } }$$

Evaluate

$$\frac{\sqrt{70}+4\sqrt{5}}{10}\approx 1.731087218$$

Solution Steps

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{\sqrt{7}+2\sqrt{2}}{\sqrt{10}}$$

Rationalize the denominator of \frac{\sqrt{7}+2\sqrt{2}}{\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{\left(\sqrt{7}+2\sqrt{2}\right)\sqrt{10}}{\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{\left(\sqrt{7}+2\sqrt{2}\right)\sqrt{10}}{10}$$

Use the distributive property to multiply \sqrt{7}+2\sqrt{2} by \sqrt{10}.

$$\frac{\sqrt{7}\sqrt{10}+2\sqrt{2}\sqrt{10}}{10}$$

To multiply \sqrt{7} and \sqrt{10}, multiply the numbers under the square root.

$$\frac{\sqrt{70}+2\sqrt{2}\sqrt{10}}{10}$$

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}.

$$\frac{\sqrt{70}+2\sqrt{2}\sqrt{2}\sqrt{5}}{10}$$

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

$$\frac{\sqrt{70}+2\times 2\sqrt{5}}{10}$$

Multiply 2 and 2 to get 4.

$$\frac{\sqrt{70}+4\sqrt{5}}{10}$$