Solve sqrt{31/3}div(2/5sqrt{1/3})*(4sqrt{2/5})

$$\sqrt { 3 \frac { 1 } { 3 } } \div ( \frac { 2 } { 5 } \sqrt { \frac { 1 } { 3 } } ) \times ( 4 \sqrt { \frac { 2 } { 5 } } )$$

Evaluate

$$20$$

Solution Steps

Multiply 3 and 3 to get 9.

$$\frac{\sqrt{\frac{9+1}{3}}}{\frac{2}{5}\sqrt{\frac{1}{3}}}\times 4\sqrt{\frac{2}{5}}$$

Add 9 and 1 to get 10.

$$\frac{\sqrt{\frac{10}{3}}}{\frac{2}{5}\sqrt{\frac{1}{3}}}\times 4\sqrt{\frac{2}{5}}$$

Rewrite the square root of the division \sqrt{\frac{10}{3}} as the division of square roots \frac{\sqrt{10}}{\sqrt{3}}.

$$\frac{\frac{\sqrt{10}}{\sqrt{3}}}{\frac{2}{5}\sqrt{\frac{1}{3}}}\times 4\sqrt{\frac{2}{5}}$$

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

$$\frac{\frac{\sqrt{10}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}{\frac{2}{5}\sqrt{\frac{1}{3}}}\times 4\sqrt{\frac{2}{5}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{\sqrt{10}\sqrt{3}}{3}}{\frac{2}{5}\sqrt{\frac{1}{3}}}\times 4\sqrt{\frac{2}{5}}$$

To multiply \sqrt{10} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\frac{\sqrt{30}}{3}}{\frac{2}{5}\sqrt{\frac{1}{3}}}\times 4\sqrt{\frac{2}{5}}$$

Rewrite the square root of the division \sqrt{\frac{1}{3}} as the division of square roots \frac{\sqrt{1}}{\sqrt{3}}.

$$\frac{\frac{\sqrt{30}}{3}}{\frac{2}{5}\times \frac{\sqrt{1}}{\sqrt{3}}}\times 4\sqrt{\frac{2}{5}}$$

Calculate the square root of 1 and get 1.

$$\frac{\frac{\sqrt{30}}{3}}{\frac{2}{5}\times \frac{1}{\sqrt{3}}}\times 4\sqrt{\frac{2}{5}}$$

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\frac{\sqrt{30}}{3}}{\frac{2}{5}\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}\times 4\sqrt{\frac{2}{5}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{\sqrt{30}}{3}}{\frac{2}{5}\times \frac{\sqrt{3}}{3}}\times 4\sqrt{\frac{2}{5}}$$

Multiply \frac{2}{5} times \frac{\sqrt{3}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\sqrt{30}}{3}}{\frac{2\sqrt{3}}{5\times 3}}\times 4\sqrt{\frac{2}{5}}$$

Divide \frac{\sqrt{30}}{3} by \frac{2\sqrt{3}}{5\times 3} by multiplying \frac{\sqrt{30}}{3} by the reciprocal of \frac{2\sqrt{3}}{5\times 3}.

$$\frac{\sqrt{30}\times 5\times 3}{3\times 2\sqrt{3}}\times 4\sqrt{\frac{2}{5}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{5\sqrt{30}}{2\sqrt{3}}\times 4\sqrt{\frac{2}{5}}$$

Rationalize the denominator of \frac{5\sqrt{30}}{2\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{5\sqrt{30}\sqrt{3}}{2\left(\sqrt{3}\right)^{2}}\times 4\sqrt{\frac{2}{5}}$$

The square of \sqrt{3} is 3.

$$\frac{5\sqrt{30}\sqrt{3}}{2\times 3}\times 4\sqrt{\frac{2}{5}}$$

Factor 30=3\times 10. Rewrite the square root of the product \sqrt{3\times 10} as the product of square roots \sqrt{3}\sqrt{10}.

$$\frac{5\sqrt{3}\sqrt{10}\sqrt{3}}{2\times 3}\times 4\sqrt{\frac{2}{5}}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{5\times 3\sqrt{10}}{2\times 3}\times 4\sqrt{\frac{2}{5}}$$

Multiply 2 and 3 to get 6.

$$\frac{5\times 3\sqrt{10}}{6}\times 4\sqrt{\frac{2}{5}}$$

Multiply 5 and 3 to get 15.

$$\frac{15\sqrt{10}}{6}\times 4\sqrt{\frac{2}{5}}$$

Divide 15\sqrt{10} by 6 to get \frac{5}{2}\sqrt{10}.

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

Express \frac{5}{2}\times 4 as a single fraction.

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

Multiply 5 and 4 to get 20.

$$\frac{20}{2}\sqrt{10}\sqrt{\frac{2}{5}}$$

Divide 20 by 2 to get 10.

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

Rewrite the square root of the division \sqrt{\frac{2}{5}} as the division of square roots \frac{\sqrt{2}}{\sqrt{5}}.

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

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

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

The square of \sqrt{5} is 5.

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

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

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

Cancel out 5, the greatest common factor in 10 and 5.

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

Multiply \sqrt{10} and \sqrt{10} to get 10.

$$2\times 10$$

Multiply 2 and 10 to get 20.

$$20$$

Factor

$$2^{2}\times 5$$