Solve sqrt{22/3}divsqrt{21/3}sqrt{12/5}

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

Evaluate

$$\frac{2\sqrt{10}}{5}\approx 1.264911064$$

Solution Steps

Multiply 2 and 3 to get 6.

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

Add 6 and 2 to get 8.

$$\frac{\sqrt{\frac{8}{3}}}{\sqrt{\frac{2\times 3+1}{3}}}\sqrt{\frac{1\times 5+2}{5}}$$

Rewrite the square root of the division \sqrt{\frac{8}{3}} as the division of square roots \frac{\sqrt{8}}{\sqrt{3}}.

$$\frac{\frac{\sqrt{8}}{\sqrt{3}}}{\sqrt{\frac{2\times 3+1}{3}}}\sqrt{\frac{1\times 5+2}{5}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

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

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

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

The square of \sqrt{3} is 3.

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

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

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

Multiply 2 and 3 to get 6.

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

Add 6 and 1 to get 7.

$$\frac{\frac{2\sqrt{6}}{3}}{\sqrt{\frac{7}{3}}}\sqrt{\frac{1\times 5+2}{5}}$$

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

$$\frac{\frac{2\sqrt{6}}{3}}{\frac{\sqrt{7}}{\sqrt{3}}}\sqrt{\frac{1\times 5+2}{5}}$$

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

$$\frac{\frac{2\sqrt{6}}{3}}{\frac{\sqrt{7}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}\sqrt{\frac{1\times 5+2}{5}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{2\sqrt{6}}{3}}{\frac{\sqrt{7}\sqrt{3}}{3}}\sqrt{\frac{1\times 5+2}{5}}$$

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

$$\frac{\frac{2\sqrt{6}}{3}}{\frac{\sqrt{21}}{3}}\sqrt{\frac{1\times 5+2}{5}}$$

Divide \frac{2\sqrt{6}}{3} by \frac{\sqrt{21}}{3} by multiplying \frac{2\sqrt{6}}{3} by the reciprocal of \frac{\sqrt{21}}{3}.

$$\frac{2\sqrt{6}\times 3}{3\sqrt{21}}\sqrt{\frac{1\times 5+2}{5}}$$

Cancel out 3 in both numerator and denominator.

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

Rationalize the denominator of \frac{2\sqrt{6}}{\sqrt{21}} by multiplying numerator and denominator by \sqrt{21}.

$$\frac{2\sqrt{6}\sqrt{21}}{\left(\sqrt{21}\right)^{2}}\sqrt{\frac{1\times 5+2}{5}}$$

The square of \sqrt{21} is 21.

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

To multiply \sqrt{6} and \sqrt{21}, multiply the numbers under the square root.

$$\frac{2\sqrt{126}}{21}\sqrt{\frac{1\times 5+2}{5}}$$

Factor 126=3^{2}\times 14. Rewrite the square root of the product \sqrt{3^{2}\times 14} as the product of square roots \sqrt{3^{2}}\sqrt{14}. Take the square root of 3^{2}.

$$\frac{2\times 3\sqrt{14}}{21}\sqrt{\frac{1\times 5+2}{5}}$$

Multiply 2 and 3 to get 6.

$$\frac{6\sqrt{14}}{21}\sqrt{\frac{1\times 5+2}{5}}$$

Divide 6\sqrt{14} by 21 to get \frac{2}{7}\sqrt{14}.

$$\frac{2}{7}\sqrt{14}\sqrt{\frac{1\times 5+2}{5}}$$

Multiply 1 and 5 to get 5.

$$\frac{2}{7}\sqrt{14}\sqrt{\frac{5+2}{5}}$$

Add 5 and 2 to get 7.

$$\frac{2}{7}\sqrt{14}\sqrt{\frac{7}{5}}$$

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

$$\frac{2}{7}\sqrt{14}\times \frac{\sqrt{7}}{\sqrt{5}}$$

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

$$\frac{2}{7}\sqrt{14}\times \frac{\sqrt{7}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{2}{7}\sqrt{14}\times \frac{\sqrt{7}\sqrt{5}}{5}$$

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

$$\frac{2}{7}\sqrt{14}\times \frac{\sqrt{35}}{5}$$

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

$$\frac{2\sqrt{35}}{7\times 5}\sqrt{14}$$

Multiply 7 and 5 to get 35.

$$\frac{2\sqrt{35}}{35}\sqrt{14}$$

Express \frac{2\sqrt{35}}{35}\sqrt{14} as a single fraction.

$$\frac{2\sqrt{35}\sqrt{14}}{35}$$

To multiply \sqrt{35} and \sqrt{14}, multiply the numbers under the square root.

$$\frac{2\sqrt{490}}{35}$$

Factor 490=7^{2}\times 10. Rewrite the square root of the product \sqrt{7^{2}\times 10} as the product of square roots \sqrt{7^{2}}\sqrt{10}. Take the square root of 7^{2}.

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

Multiply 2 and 7 to get 14.

$$\frac{14\sqrt{10}}{35}$$

Divide 14\sqrt{10} by 35 to get \frac{2}{5}\sqrt{10}.

$$\frac{2}{5}\sqrt{10}$$