Solve sqrt{31/3}div2/5sqrt{21/3}*4sqrt{2/5}

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

Evaluate

$$\frac{20\sqrt{7}}{3}\approx 17.638342074$$

Solution Steps

Multiply 3 and 3 to get 9.

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

Add 9 and 1 to get 10.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

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

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

Multiply 3 and 2 to get 6.

$$4\times \frac{\sqrt{30}\times 5}{6}\sqrt{\frac{2\times 3+1}{3}}\sqrt{\frac{2}{5}}$$

Multiply 2 and 3 to get 6.

$$4\times \frac{\sqrt{30}\times 5}{6}\sqrt{\frac{6+1}{3}}\sqrt{\frac{2}{5}}$$

Add 6 and 1 to get 7.

$$4\times \frac{\sqrt{30}\times 5}{6}\sqrt{\frac{7}{3}}\sqrt{\frac{2}{5}}$$

Rewrite the square root of the division \sqrt{\frac{7}{3}} as the division of square roots \frac{\sqrt{7}}{\sqrt{3}}.

$$4\times \frac{\sqrt{30}\times 5}{6}\times \frac{\sqrt{7}}{\sqrt{3}}\sqrt{\frac{2}{5}}$$

Rationalize the denominator of \frac{\sqrt{7}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$4\times \frac{\sqrt{30}\times 5}{6}\times \frac{\sqrt{7}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\sqrt{\frac{2}{5}}$$

The square of \sqrt{3} is 3.

$$4\times \frac{\sqrt{30}\times 5}{6}\times \frac{\sqrt{7}\sqrt{3}}{3}\sqrt{\frac{2}{5}}$$

To multiply \sqrt{7} and \sqrt{3}, multiply the numbers under the square root.

$$4\times \frac{\sqrt{30}\times 5}{6}\times \frac{\sqrt{21}}{3}\sqrt{\frac{2}{5}}$$

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

$$4\times \frac{\sqrt{30}\times 5}{6}\times \frac{\sqrt{21}}{3}\times \frac{\sqrt{2}}{\sqrt{5}}$$

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

$$4\times \frac{\sqrt{30}\times 5}{6}\times \frac{\sqrt{21}}{3}\times \frac{\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$4\times \frac{\sqrt{30}\times 5}{6}\times \frac{\sqrt{21}}{3}\times \frac{\sqrt{2}\sqrt{5}}{5}$$

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

$$4\times \frac{\sqrt{30}\times 5}{6}\times \frac{\sqrt{21}}{3}\times \frac{\sqrt{10}}{5}$$

Express 4\times \frac{\sqrt{30}\times 5}{6} as a single fraction.

$$\frac{4\sqrt{30}\times 5}{6}\times \frac{\sqrt{21}}{3}\times \frac{\sqrt{10}}{5}$$

Multiply \frac{4\sqrt{30}\times 5}{6} times \frac{\sqrt{21}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4\sqrt{30}\times 5\sqrt{21}}{6\times 3}\times \frac{\sqrt{10}}{5}$$

Cancel out 2 in both numerator and denominator.

$$\frac{2\times 5\sqrt{21}\sqrt{30}}{3\times 3}\times \frac{\sqrt{10}}{5}$$

Multiply \frac{2\times 5\sqrt{21}\sqrt{30}}{3\times 3} times \frac{\sqrt{10}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\times 5\sqrt{21}\sqrt{30}\sqrt{10}}{3\times 3\times 5}$$

Cancel out 5 in both numerator and denominator.

$$\frac{2\sqrt{10}\sqrt{21}\sqrt{30}}{3\times 3}$$

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}.

$$\frac{2\sqrt{10}\sqrt{21}\sqrt{10}\sqrt{3}}{3\times 3}$$

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

$$\frac{2\times 10\sqrt{21}\sqrt{3}}{3\times 3}$$

Factor 21=3\times 7. Rewrite the square root of the product \sqrt{3\times 7} as the product of square roots \sqrt{3}\sqrt{7}.

$$\frac{2\times 10\sqrt{3}\sqrt{7}\sqrt{3}}{3\times 3}$$

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

$$\frac{2\times 10\times 3\sqrt{7}}{3\times 3}$$

Multiply 2 and 10 to get 20.

$$\frac{20\times 3\sqrt{7}}{3\times 3}$$

Multiply 20 and 3 to get 60.

$$\frac{60\sqrt{7}}{3\times 3}$$

Multiply 3 and 3 to get 9.

$$\frac{60\sqrt{7}}{9}$$

Divide 60\sqrt{7} by 9 to get \frac{20}{3}\sqrt{7}.

$$\frac{20}{3}\sqrt{7}$$