Solve sqrt{4/5}divsqrt{24/5}*sqrt{7/10}

$$\sqrt { \frac { 4 } { 5 } } \div \sqrt { 2 \frac { 4 } { 5 } } \times \sqrt { \frac { 7 } { 10 } }$$

Evaluate

$$\frac{\sqrt{5}}{5}\approx 0.447213595$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{4}{5}} as the division of square roots \frac{\sqrt{4}}{\sqrt{5}}.

$$\frac{\frac{\sqrt{4}}{\sqrt{5}}}{\sqrt{\frac{2\times 5+4}{5}}}\sqrt{\frac{7}{10}}$$

Calculate the square root of 4 and get 2.

$$\frac{\frac{2}{\sqrt{5}}}{\sqrt{\frac{2\times 5+4}{5}}}\sqrt{\frac{7}{10}}$$

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

$$\frac{\frac{2\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}{\sqrt{\frac{2\times 5+4}{5}}}\sqrt{\frac{7}{10}}$$

The square of \sqrt{5} is 5.

$$\frac{\frac{2\sqrt{5}}{5}}{\sqrt{\frac{2\times 5+4}{5}}}\sqrt{\frac{7}{10}}$$

Multiply 2 and 5 to get 10.

$$\frac{\frac{2\sqrt{5}}{5}}{\sqrt{\frac{10+4}{5}}}\sqrt{\frac{7}{10}}$$

Add 10 and 4 to get 14.

$$\frac{\frac{2\sqrt{5}}{5}}{\sqrt{\frac{14}{5}}}\sqrt{\frac{7}{10}}$$

Rewrite the square root of the division \sqrt{\frac{14}{5}} as the division of square roots \frac{\sqrt{14}}{\sqrt{5}}.

$$\frac{\frac{2\sqrt{5}}{5}}{\frac{\sqrt{14}}{\sqrt{5}}}\sqrt{\frac{7}{10}}$$

Rationalize the denominator of \frac{\sqrt{14}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\frac{2\sqrt{5}}{5}}{\frac{\sqrt{14}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}\sqrt{\frac{7}{10}}$$

The square of \sqrt{5} is 5.

$$\frac{\frac{2\sqrt{5}}{5}}{\frac{\sqrt{14}\sqrt{5}}{5}}\sqrt{\frac{7}{10}}$$

To multiply \sqrt{14} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\frac{2\sqrt{5}}{5}}{\frac{\sqrt{70}}{5}}\sqrt{\frac{7}{10}}$$

Divide \frac{2\sqrt{5}}{5} by \frac{\sqrt{70}}{5} by multiplying \frac{2\sqrt{5}}{5} by the reciprocal of \frac{\sqrt{70}}{5}.

$$\frac{2\sqrt{5}\times 5}{5\sqrt{70}}\sqrt{\frac{7}{10}}$$

Cancel out 5 in both numerator and denominator.

$$\frac{2\sqrt{5}}{\sqrt{70}}\sqrt{\frac{7}{10}}$$

Rationalize the denominator of \frac{2\sqrt{5}}{\sqrt{70}} by multiplying numerator and denominator by \sqrt{70}.

$$\frac{2\sqrt{5}\sqrt{70}}{\left(\sqrt{70}\right)^{2}}\sqrt{\frac{7}{10}}$$

The square of \sqrt{70} is 70.

$$\frac{2\sqrt{5}\sqrt{70}}{70}\sqrt{\frac{7}{10}}$$

Factor 70=5\times 14. Rewrite the square root of the product \sqrt{5\times 14} as the product of square roots \sqrt{5}\sqrt{14}.

$$\frac{2\sqrt{5}\sqrt{5}\sqrt{14}}{70}\sqrt{\frac{7}{10}}$$

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

$$\frac{2\times 5\sqrt{14}}{70}\sqrt{\frac{7}{10}}$$

Multiply 2 and 5 to get 10.

$$\frac{10\sqrt{14}}{70}\sqrt{\frac{7}{10}}$$

Divide 10\sqrt{14} by 70 to get \frac{1}{7}\sqrt{14}.

$$\frac{1}{7}\sqrt{14}\sqrt{\frac{7}{10}}$$

Rewrite the square root of the division \sqrt{\frac{7}{10}} as the division of square roots \frac{\sqrt{7}}{\sqrt{10}}.

$$\frac{1}{7}\sqrt{14}\times \frac{\sqrt{7}}{\sqrt{10}}$$

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

$$\frac{1}{7}\sqrt{14}\times \frac{\sqrt{7}\sqrt{10}}{\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{1}{7}\sqrt{14}\times \frac{\sqrt{7}\sqrt{10}}{10}$$

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

$$\frac{1}{7}\sqrt{14}\times \frac{\sqrt{70}}{10}$$

Multiply \frac{1}{7} times \frac{\sqrt{70}}{10} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{70}}{7\times 10}\sqrt{14}$$

Multiply 7 and 10 to get 70.

$$\frac{\sqrt{70}}{70}\sqrt{14}$$

Express \frac{\sqrt{70}}{70}\sqrt{14} as a single fraction.

$$\frac{\sqrt{70}\sqrt{14}}{70}$$

Factor 70=14\times 5. Rewrite the square root of the product \sqrt{14\times 5} as the product of square roots \sqrt{14}\sqrt{5}.

$$\frac{\sqrt{14}\sqrt{5}\sqrt{14}}{70}$$

Multiply \sqrt{14} and \sqrt{14} to get 14.

$$\frac{14\sqrt{5}}{70}$$

Divide 14\sqrt{5} by 70 to get \frac{1}{5}\sqrt{5}.

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