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

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

Evaluate

$$3\sqrt{2}x$$

Short Solution Steps

Multiply 3 and 3 to get 9.

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

Add 9 and 1 to get 10.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

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

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

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

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

The square of \sqrt{30} is 30.

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

Multiply 2 and 3 to get 6.

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

Add 6 and 1 to get 7.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

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

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

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

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

Cancel out 3\times 5 in both numerator and denominator.

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

Rationalize the denominator of \frac{3\sqrt{30}x}{2\times 2\sqrt{21}} by multiplying numerator and denominator by \sqrt{21}.

$$\frac{3\sqrt{30}x\sqrt{21}}{2\times 2\left(\sqrt{21}\right)^{2}}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

The square of \sqrt{21} is 21.

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

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

$$\frac{3\sqrt{630}x}{2\times 2\times 21}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

Multiply 2 and 2 to get 4.

$$\frac{3\sqrt{630}x}{4\times 21}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

Multiply 4 and 21 to get 84.

$$\frac{3\sqrt{630}x}{84}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

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}.

$$\frac{3\times 3\sqrt{70}x}{84}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

Multiply 3 and 3 to get 9.

$$\frac{9\sqrt{70}x}{84}\times 4\sqrt{\frac{1\times 5+2}{5}}$$

Divide 9\sqrt{70}x by 84 to get \frac{3}{28}\sqrt{70}x.

$$\frac{3}{28}\sqrt{70}x\times 4\sqrt{\frac{1\times 5+2}{5}}$$

Express \frac{3}{28}\times 4 as a single fraction.

$$\frac{3\times 4}{28}\sqrt{70}x\sqrt{\frac{1\times 5+2}{5}}$$

Multiply 3 and 4 to get 12.

$$\frac{12}{28}\sqrt{70}x\sqrt{\frac{1\times 5+2}{5}}$$

Reduce the fraction \frac{12}{28} to lowest terms by extracting and canceling out 4.

$$\frac{3}{7}\sqrt{70}x\sqrt{\frac{1\times 5+2}{5}}$$

Multiply 1 and 5 to get 5.

$$\frac{3}{7}\sqrt{70}x\sqrt{\frac{5+2}{5}}$$

Add 5 and 2 to get 7.

$$\frac{3}{7}\sqrt{70}x\sqrt{\frac{7}{5}}$$

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

$$\frac{3}{7}\sqrt{70}x\times \frac{\sqrt{7}}{\sqrt{5}}$$

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

$$\frac{3}{7}\sqrt{70}x\times \frac{\sqrt{7}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{3}{7}\sqrt{70}x\times \frac{\sqrt{7}\sqrt{5}}{5}$$

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

$$\frac{3}{7}\sqrt{70}x\times \frac{\sqrt{35}}{5}$$

Multiply \frac{3}{7} times \frac{\sqrt{35}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\sqrt{35}}{7\times 5}\sqrt{70}x$$

Multiply 7 and 5 to get 35.

$$\frac{3\sqrt{35}}{35}\sqrt{70}x$$

Express \frac{3\sqrt{35}}{35}\sqrt{70} as a single fraction.

$$\frac{3\sqrt{35}\sqrt{70}}{35}x$$

Express \frac{3\sqrt{35}\sqrt{70}}{35}x as a single fraction.

$$\frac{3\sqrt{35}\sqrt{70}x}{35}$$

Differentiate w.r.t. x

$$3 \sqrt{2} = 4.242640687$$