Solve sqrt{31/5}divsqrt{13/5}

$$\sqrt { 3 \frac { 1 } { 5 } } \div \sqrt { 1 \frac { 3 } { 5 } }$$

Evaluate

$$\sqrt{2}\approx 1.414213562$$

Solution Steps

Multiply 3 and 5 to get 15.

$$\frac{\sqrt{\frac{15+1}{5}}}{\sqrt{\frac{1\times 5+3}{5}}}$$

Add 15 and 1 to get 16.

$$\frac{\sqrt{\frac{16}{5}}}{\sqrt{\frac{1\times 5+3}{5}}}$$

Rewrite the square root of the division \sqrt{\frac{16}{5}} as the division of square roots \frac{\sqrt{16}}{\sqrt{5}}.

$$\frac{\frac{\sqrt{16}}{\sqrt{5}}}{\sqrt{\frac{1\times 5+3}{5}}}$$

Calculate the square root of 16 and get 4.

$$\frac{\frac{4}{\sqrt{5}}}{\sqrt{\frac{1\times 5+3}{5}}}$$

Rationalize the denominator of \frac{4}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\frac{4\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}{\sqrt{\frac{1\times 5+3}{5}}}$$

The square of \sqrt{5} is 5.

$$\frac{\frac{4\sqrt{5}}{5}}{\sqrt{\frac{1\times 5+3}{5}}}$$

Multiply 1 and 5 to get 5.

$$\frac{\frac{4\sqrt{5}}{5}}{\sqrt{\frac{5+3}{5}}}$$

Add 5 and 3 to get 8.

$$\frac{\frac{4\sqrt{5}}{5}}{\sqrt{\frac{8}{5}}}$$

Rewrite the square root of the division \sqrt{\frac{8}{5}} as the division of square roots \frac{\sqrt{8}}{\sqrt{5}}.

$$\frac{\frac{4\sqrt{5}}{5}}{\frac{\sqrt{8}}{\sqrt{5}}}$$

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}.

$$\frac{\frac{4\sqrt{5}}{5}}{\frac{2\sqrt{2}}{\sqrt{5}}}$$

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

$$\frac{\frac{4\sqrt{5}}{5}}{\frac{2\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}$$

The square of \sqrt{5} is 5.

$$\frac{\frac{4\sqrt{5}}{5}}{\frac{2\sqrt{2}\sqrt{5}}{5}}$$

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

$$\frac{\frac{4\sqrt{5}}{5}}{\frac{2\sqrt{10}}{5}}$$

Divide \frac{4\sqrt{5}}{5} by \frac{2\sqrt{10}}{5} by multiplying \frac{4\sqrt{5}}{5} by the reciprocal of \frac{2\sqrt{10}}{5}.

$$\frac{4\sqrt{5}\times 5}{5\times 2\sqrt{10}}$$

Cancel out 2\times 5 in both numerator and denominator.

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

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

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

The square of \sqrt{10} is 10.

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

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}.

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

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

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

Multiply 2 and 5 to get 10.

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

Cancel out 10 and 10.

$$\sqrt{2}$$