Solve x=sqrt{31/3}div(2/5sqrt{21/3})*(4sqrt{12/5})

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

Solve for x

$$x=10\sqrt{2}\approx 14.142135624$$

Steps for Solving Linear Equation

Multiply 3 and 3 to get 9.

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

Add 9 and 1 to get 10.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

Multiply 2 and 3 to get 6.

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

Add 6 and 1 to get 7.

$$x=\frac{\frac{\sqrt{30}}{3}}{\frac{2}{5}\sqrt{\frac{7}{3}}}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

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

$$x=\frac{\frac{\sqrt{30}}{3}}{\frac{2}{5}\times \frac{\sqrt{7}}{\sqrt{3}}}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

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

$$x=\frac{\frac{\sqrt{30}}{3}}{\frac{2}{5}\times \frac{\sqrt{7}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

The square of \sqrt{3} is 3.

$$x=\frac{\frac{\sqrt{30}}{3}}{\frac{2}{5}\times \frac{\sqrt{7}\sqrt{3}}{3}}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

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

$$x=\frac{\frac{\sqrt{30}}{3}}{\frac{2}{5}\times \frac{\sqrt{21}}{3}}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

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

$$x=\frac{\frac{\sqrt{30}}{3}}{\frac{2\sqrt{21}}{5\times 3}}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

Divide \frac{\sqrt{30}}{3} by \frac{2\sqrt{21}}{5\times 3} by multiplying \frac{\sqrt{30}}{3} by the reciprocal of \frac{2\sqrt{21}}{5\times 3}.

$$x=\frac{\sqrt{30}\times 5\times 3}{3\times 2\sqrt{21}}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

Cancel out 3 in both numerator and denominator.

$$x=\frac{5\sqrt{30}}{2\sqrt{21}}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

Rationalize the denominator of \frac{5\sqrt{30}}{2\sqrt{21}} by multiplying numerator and denominator by \sqrt{21}.

$$x=\frac{5\sqrt{30}\sqrt{21}}{2\left(\sqrt{21}\right)^{2}}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

The square of \sqrt{21} is 21.

$$x=\frac{5\sqrt{30}\sqrt{21}}{2\times 21}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

To multiply \sqrt{30} and \sqrt{21}, multiply the numbers under the square root.

$$x=\frac{5\sqrt{630}}{2\times 21}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

Multiply 2 and 21 to get 42.

$$x=\frac{5\sqrt{630}}{42}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

Factor 630=3^{2}\times 70. Rewrite the square root of the product \sqrt{3^{2}\times 70} as the product of square roots \sqrt{3^{2}}\sqrt{70}. Take the square root of 3^{2}.

$$x=\frac{5\times 3\sqrt{70}}{42}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

Multiply 5 and 3 to get 15.

$$x=\frac{15\sqrt{70}}{42}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

Divide 15\sqrt{70} by 42 to get \frac{5}{14}\sqrt{70}.

$$x=\frac{5}{14}\sqrt{70}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

Express \frac{5}{14}\times 4 as a single fraction.

$$x=\frac{5\times 4}{14}\sqrt{70}\sqrt{\frac{1\times 5+2}{5}}$$

Multiply 5 and 4 to get 20.

$$x=\frac{20}{14}\sqrt{70}\sqrt{\frac{1\times 5+2}{5}}$$

Reduce the fraction \frac{20}{14} to lowest terms by extracting and canceling out 2.

$$x=\frac{10}{7}\sqrt{70}\sqrt{\frac{1\times 5+2}{5}}$$

Multiply 1 and 5 to get 5.

$$x=\frac{10}{7}\sqrt{70}\sqrt{\frac{5+2}{5}}$$

Add 5 and 2 to get 7.

$$x=\frac{10}{7}\sqrt{70}\sqrt{\frac{7}{5}}$$

Rewrite the square root of the division \sqrt{\frac{7}{5}} as the division of square roots \frac{\sqrt{7}}{\sqrt{5}}.

$$x=\frac{10}{7}\sqrt{70}\times \frac{\sqrt{7}}{\sqrt{5}}$$

Rationalize the denominator of \frac{\sqrt{7}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

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

The square of \sqrt{5} is 5.

$$x=\frac{10}{7}\sqrt{70}\times \frac{\sqrt{7}\sqrt{5}}{5}$$

To multiply \sqrt{7} and \sqrt{5}, multiply the numbers under the square root.

$$x=\frac{10}{7}\sqrt{70}\times \frac{\sqrt{35}}{5}$$

Multiply \frac{10}{7} times \frac{\sqrt{35}}{5} by multiplying numerator times numerator and denominator times denominator.

$$x=\frac{10\sqrt{35}}{7\times 5}\sqrt{70}$$

Cancel out 5 in both numerator and denominator.

$$x=\frac{2\sqrt{35}}{7}\sqrt{70}$$

Express \frac{2\sqrt{35}}{7}\sqrt{70} as a single fraction.

$$x=\frac{2\sqrt{35}\sqrt{70}}{7}$$

Factor 70=35\times 2. Rewrite the square root of the product \sqrt{35\times 2} as the product of square roots \sqrt{35}\sqrt{2}.

$$x=\frac{2\sqrt{35}\sqrt{35}\sqrt{2}}{7}$$

Multiply \sqrt{35} and \sqrt{35} to get 35.

$$x=\frac{2\times 35\sqrt{2}}{7}$$

Multiply 2 and 35 to get 70.

$$x=\frac{70\sqrt{2}}{7}$$

Divide 70\sqrt{2} by 7 to get 10\sqrt{2}.

$$x=10\sqrt{2}$$

Assign x

$$x≔10\sqrt{2}$$