Solve sqrt{11/3}divsqrt{21/3}divsqrt{12/5}

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

Evaluate

$$\frac{2\sqrt{5}}{7}\approx 0.638876565$$

Solution Steps

Multiply 1 and 3 to get 3.

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

Add 3 and 1 to get 4.

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

Rewrite the square root of the division \sqrt{\frac{4}{3}} as the division of square roots \frac{\sqrt{4}}{\sqrt{3}}.

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

Calculate the square root of 4 and get 2.

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

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

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

The square of \sqrt{3} is 3.

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

Multiply 2 and 3 to get 6.

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

Add 6 and 1 to get 7.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

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

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

Cancel out 3 in both numerator and denominator.

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

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

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

The square of \sqrt{21} is 21.

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

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}.

$$\frac{\frac{2\sqrt{3}\sqrt{3}\sqrt{7}}{21}}{\sqrt{\frac{1\times 5+2}{5}}}$$

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

$$\frac{\frac{2\times 3\sqrt{7}}{21}}{\sqrt{\frac{1\times 5+2}{5}}}$$

Multiply 2 and 3 to get 6.

$$\frac{\frac{6\sqrt{7}}{21}}{\sqrt{\frac{1\times 5+2}{5}}}$$

Divide 6\sqrt{7} by 21 to get \frac{2}{7}\sqrt{7}.

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

Multiply 1 and 5 to get 5.

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

Add 5 and 2 to get 7.

$$\frac{\frac{2}{7}\sqrt{7}}{\sqrt{\frac{7}{5}}}$$

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

$$\frac{\frac{2}{7}\sqrt{7}}{\frac{\sqrt{7}}{\sqrt{5}}}$$

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

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

The square of \sqrt{5} is 5.

$$\frac{\frac{2}{7}\sqrt{7}}{\frac{\sqrt{7}\sqrt{5}}{5}}$$

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

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

Divide \frac{2}{7}\sqrt{7} by \frac{\sqrt{35}}{5} by multiplying \frac{2}{7}\sqrt{7} by the reciprocal of \frac{\sqrt{35}}{5}.

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

Rationalize the denominator of \frac{\frac{2}{7}\sqrt{7}\times 5}{\sqrt{35}} by multiplying numerator and denominator by \sqrt{35}.

$$\frac{\frac{2}{7}\sqrt{7}\times 5\sqrt{35}}{\left(\sqrt{35}\right)^{2}}$$

The square of \sqrt{35} is 35.

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

Express \frac{2}{7}\times 5 as a single fraction.

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

Multiply 2 and 5 to get 10.

$$\frac{\frac{10}{7}\sqrt{7}\sqrt{35}}{35}$$

Factor 35=7\times 5. Rewrite the square root of the product \sqrt{7\times 5} as the product of square roots \sqrt{7}\sqrt{5}.

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

Multiply \sqrt{7} and \sqrt{7} to get 7.

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

Cancel out 7 and 7.

$$\frac{10\sqrt{5}}{35}$$

Divide 10\sqrt{5} by 35 to get \frac{2}{7}\sqrt{5}.

$$\frac{2}{7}\sqrt{5}$$