Solve sqrt{31/3}div(2/5sqrt{21/2})*4sqrt{2/5})

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

Evaluate

$$\frac{4\sqrt{30}}{3}\approx 7.302967433$$

Solution Steps

Multiply 3 and 3 to get 9.

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

Add 9 and 1 to get 10.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

Multiply 2 and 2 to get 4.

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

Add 4 and 1 to get 5.

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

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

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

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

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

The square of \sqrt{2} is 2.

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

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

$$4\times \frac{\frac{\sqrt{30}}{3}}{\frac{2}{5}\times \frac{\sqrt{10}}{2}}\sqrt{\frac{2}{5}}$$

Multiply \frac{2}{5} times \frac{\sqrt{10}}{2} by multiplying numerator times numerator and denominator times denominator.

$$4\times \frac{\frac{\sqrt{30}}{3}}{\frac{2\sqrt{10}}{5\times 2}}\sqrt{\frac{2}{5}}$$

Cancel out 2 in both numerator and denominator.

$$4\times \frac{\frac{\sqrt{30}}{3}}{\frac{\sqrt{10}}{5}}\sqrt{\frac{2}{5}}$$

Divide \frac{\sqrt{30}}{3} by \frac{\sqrt{10}}{5} by multiplying \frac{\sqrt{30}}{3} by the reciprocal of \frac{\sqrt{10}}{5}.

$$4\times \frac{\sqrt{30}\times 5}{3\sqrt{10}}\sqrt{\frac{2}{5}}$$

Rationalize the denominator of \frac{\sqrt{30}\times 5}{3\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$4\times \frac{\sqrt{30}\times 5\sqrt{10}}{3\left(\sqrt{10}\right)^{2}}\sqrt{\frac{2}{5}}$$

The square of \sqrt{10} is 10.

$$4\times \frac{\sqrt{30}\times 5\sqrt{10}}{3\times 10}\sqrt{\frac{2}{5}}$$

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

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

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

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

Multiply 3 and 10 to get 30.

$$4\times \frac{10\times 5\sqrt{3}}{30}\sqrt{\frac{2}{5}}$$

Multiply 10 and 5 to get 50.

$$4\times \frac{50\sqrt{3}}{30}\sqrt{\frac{2}{5}}$$

Divide 50\sqrt{3} by 30 to get \frac{5}{3}\sqrt{3}.

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

Express 4\times \frac{5}{3} as a single fraction.

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

Multiply 4 and 5 to get 20.

$$\frac{20}{3}\sqrt{3}\sqrt{\frac{2}{5}}$$

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

$$\frac{20}{3}\sqrt{3}\times \frac{\sqrt{2}}{\sqrt{5}}$$

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

$$\frac{20}{3}\sqrt{3}\times \frac{\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{20}{3}\sqrt{3}\times \frac{\sqrt{2}\sqrt{5}}{5}$$

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

$$\frac{20}{3}\sqrt{3}\times \frac{\sqrt{10}}{5}$$

Multiply \frac{20}{3} times \frac{\sqrt{10}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{20\sqrt{10}}{3\times 5}\sqrt{3}$$

Cancel out 5 in both numerator and denominator.

$$\frac{4\sqrt{10}}{3}\sqrt{3}$$

Express \frac{4\sqrt{10}}{3}\sqrt{3} as a single fraction.

$$\frac{4\sqrt{10}\sqrt{3}}{3}$$

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

$$\frac{4\sqrt{30}}{3}$$