Solve sqrt{1/10}divsqrt{1/5}

$$\sqrt { \frac { 1 } { 10 } } \div \sqrt { \frac { 1 } { 5 } }$$

Evaluate

$$\frac{\sqrt{2}}{2}\approx 0.707106781$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{1}{10}} as the division of square roots \frac{\sqrt{1}}{\sqrt{10}}.

$$\frac{\frac{\sqrt{1}}{\sqrt{10}}}{\sqrt{\frac{1}{5}}}$$

Calculate the square root of 1 and get 1.

$$\frac{\frac{1}{\sqrt{10}}}{\sqrt{\frac{1}{5}}}$$

Rationalize the denominator of \frac{1}{\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{\frac{\sqrt{10}}{\left(\sqrt{10}\right)^{2}}}{\sqrt{\frac{1}{5}}}$$

The square of \sqrt{10} is 10.

$$\frac{\frac{\sqrt{10}}{10}}{\sqrt{\frac{1}{5}}}$$

Rewrite the square root of the division \sqrt{\frac{1}{5}} as the division of square roots \frac{\sqrt{1}}{\sqrt{5}}.

$$\frac{\frac{\sqrt{10}}{10}}{\frac{\sqrt{1}}{\sqrt{5}}}$$

Calculate the square root of 1 and get 1.

$$\frac{\frac{\sqrt{10}}{10}}{\frac{1}{\sqrt{5}}}$$

Rationalize the denominator of \frac{1}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

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

The square of \sqrt{5} is 5.

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

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

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

Cancel out 5 in both numerator and denominator.

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

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

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

The square of \sqrt{5} is 5.

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

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}.

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

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

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

Multiply 2 and 5 to get 10.

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

Divide 5\sqrt{2} by 10 to get \frac{1}{2}\sqrt{2}.

$$\frac{1}{2}\sqrt{2}$$