Solve sqrt{31/3}div0.4divsqrt{21/3}*4sqrt{12/5}

$$\sqrt{ 3 \frac{ 1 }{ 3 } } \div 0.4 \div \sqrt{ 2 \frac{ 1 }{ 3 } } \times 4 \sqrt{ 1 \frac{ 2 }{ 5 } }$$

Evaluate

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

Solution Steps

Multiply 3 and 3 to get 9.

$$4\times \frac{\frac{\sqrt{\frac{9+1}{3}}}{0.4}}{\sqrt{\frac{2\times 3+1}{3}}}\sqrt{\frac{1\times 5+2}{5}}$$

Add 9 and 1 to get 10.

$$4\times \frac{\frac{\sqrt{\frac{10}{3}}}{0.4}}{\sqrt{\frac{2\times 3+1}{3}}}\sqrt{\frac{1\times 5+2}{5}}$$

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

$$4\times \frac{\frac{\frac{\sqrt{10}}{\sqrt{3}}}{0.4}}{\sqrt{\frac{2\times 3+1}{3}}}\sqrt{\frac{1\times 5+2}{5}}$$

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

$$4\times \frac{\frac{\frac{\sqrt{10}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}{0.4}}{\sqrt{\frac{2\times 3+1}{3}}}\sqrt{\frac{1\times 5+2}{5}}$$

The square of \sqrt{3} is 3.

$$4\times \frac{\frac{\frac{\sqrt{10}\sqrt{3}}{3}}{0.4}}{\sqrt{\frac{2\times 3+1}{3}}}\sqrt{\frac{1\times 5+2}{5}}$$

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

$$4\times \frac{\frac{\frac{\sqrt{30}}{3}}{0.4}}{\sqrt{\frac{2\times 3+1}{3}}}\sqrt{\frac{1\times 5+2}{5}}$$

Express \frac{\frac{\sqrt{30}}{3}}{0.4} as a single fraction.

$$4\times \frac{\frac{\sqrt{30}}{3\times 0.4}}{\sqrt{\frac{2\times 3+1}{3}}}\sqrt{\frac{1\times 5+2}{5}}$$

Multiply 3 and 0.4 to get 1.2.

$$4\times \frac{\frac{\sqrt{30}}{1.2}}{\sqrt{\frac{2\times 3+1}{3}}}\sqrt{\frac{1\times 5+2}{5}}$$

Multiply 2 and 3 to get 6.

$$4\times \frac{\frac{\sqrt{30}}{1.2}}{\sqrt{\frac{6+1}{3}}}\sqrt{\frac{1\times 5+2}{5}}$$

Add 6 and 1 to get 7.

$$4\times \frac{\frac{\sqrt{30}}{1.2}}{\sqrt{\frac{7}{3}}}\sqrt{\frac{1\times 5+2}{5}}$$

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

$$4\times \frac{\frac{\sqrt{30}}{1.2}}{\frac{\sqrt{7}}{\sqrt{3}}}\sqrt{\frac{1\times 5+2}{5}}$$

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

$$4\times \frac{\frac{\sqrt{30}}{1.2}}{\frac{\sqrt{7}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}\sqrt{\frac{1\times 5+2}{5}}$$

The square of \sqrt{3} is 3.

$$4\times \frac{\frac{\sqrt{30}}{1.2}}{\frac{\sqrt{7}\sqrt{3}}{3}}\sqrt{\frac{1\times 5+2}{5}}$$

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

$$4\times \frac{\frac{\sqrt{30}}{1.2}}{\frac{\sqrt{21}}{3}}\sqrt{\frac{1\times 5+2}{5}}$$

Divide \frac{\sqrt{30}}{1.2} by \frac{\sqrt{21}}{3} by multiplying \frac{\sqrt{30}}{1.2} by the reciprocal of \frac{\sqrt{21}}{3}.

$$4\times \frac{\sqrt{30}\times 3}{1.2\sqrt{21}}\sqrt{\frac{1\times 5+2}{5}}$$

Rationalize the denominator of \frac{\sqrt{30}\times 3}{1.2\sqrt{21}} by multiplying numerator and denominator by \sqrt{21}.

$$4\times \frac{\sqrt{30}\times 3\sqrt{21}}{1.2\left(\sqrt{21}\right)^{2}}\sqrt{\frac{1\times 5+2}{5}}$$

The square of \sqrt{21} is 21.

$$4\times \frac{\sqrt{30}\times 3\sqrt{21}}{1.2\times 21}\sqrt{\frac{1\times 5+2}{5}}$$

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

$$4\times \frac{\sqrt{630}\times 3}{1.2\times 21}\sqrt{\frac{1\times 5+2}{5}}$$

Multiply 1.2 and 21 to get 25.2.

$$4\times \frac{\sqrt{630}\times 3}{25.2}\sqrt{\frac{1\times 5+2}{5}}$$

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}.

$$4\times \frac{3\sqrt{70}\times 3}{25.2}\sqrt{\frac{1\times 5+2}{5}}$$

Multiply 3 and 3 to get 9.

$$4\times \frac{9\sqrt{70}}{25.2}\sqrt{\frac{1\times 5+2}{5}}$$

Divide 9\sqrt{70} by 25.2 to get \frac{5}{14}\sqrt{70}.

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

Multiply 4 and \frac{5}{14} to get \frac{10}{7}.

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

Multiply 1 and 5 to get 5.

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

Add 5 and 2 to get 7.

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

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

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

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

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

The square of \sqrt{5} is 5.

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

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

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

Express \sqrt{70}\times \frac{\sqrt{35}}{5} as a single fraction.

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

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}.

$$\frac{10}{7}\times \frac{\sqrt{35}\sqrt{2}\sqrt{35}}{5}$$

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

$$\frac{10}{7}\times \frac{35\sqrt{2}}{5}$$

Divide 35\sqrt{2} by 5 to get 7\sqrt{2}.

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

Multiply \frac{10}{7} and 7 to get 10.

$$10\sqrt{2}$$