Solve sqrt{11/3}*sqrt{11/5}*sqrt{11/7}divsqrt{11/10}

$$\sqrt { 1 \frac { 1 } { 3 } } \times \sqrt { 1 \frac { 1 } { 5 } } \times \sqrt { 1 \frac { 1 } { 7 } } \div \sqrt { 1 \frac { 1 } { 10 } }$$

Evaluate

$$\frac{8\sqrt{154}}{77}\approx 1.289316742$$

Solution Steps

Multiply 1 and 3 to get 3.

$$\frac{\sqrt{\frac{3+1}{3}}\sqrt{\frac{1\times 5+1}{5}}\sqrt{\frac{1\times 7+1}{7}}}{\sqrt{\frac{1\times 10+1}{10}}}$$

Add 3 and 1 to get 4.

$$\frac{\sqrt{\frac{4}{3}}\sqrt{\frac{1\times 5+1}{5}}\sqrt{\frac{1\times 7+1}{7}}}{\sqrt{\frac{1\times 10+1}{10}}}$$

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

$$\frac{\frac{\sqrt{4}}{\sqrt{3}}\sqrt{\frac{1\times 5+1}{5}}\sqrt{\frac{1\times 7+1}{7}}}{\sqrt{\frac{1\times 10+1}{10}}}$$

Calculate the square root of 4 and get 2.

$$\frac{\frac{2}{\sqrt{3}}\sqrt{\frac{1\times 5+1}{5}}\sqrt{\frac{1\times 7+1}{7}}}{\sqrt{\frac{1\times 10+1}{10}}}$$

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

$$\frac{\frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\sqrt{\frac{1\times 5+1}{5}}\sqrt{\frac{1\times 7+1}{7}}}{\sqrt{\frac{1\times 10+1}{10}}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{2\sqrt{3}}{3}\sqrt{\frac{1\times 5+1}{5}}\sqrt{\frac{1\times 7+1}{7}}}{\sqrt{\frac{1\times 10+1}{10}}}$$

Multiply 1 and 5 to get 5.

$$\frac{\frac{2\sqrt{3}}{3}\sqrt{\frac{5+1}{5}}\sqrt{\frac{1\times 7+1}{7}}}{\sqrt{\frac{1\times 10+1}{10}}}$$

Add 5 and 1 to get 6.

$$\frac{\frac{2\sqrt{3}}{3}\sqrt{\frac{6}{5}}\sqrt{\frac{1\times 7+1}{7}}}{\sqrt{\frac{1\times 10+1}{10}}}$$

Rewrite the square root of the division \sqrt{\frac{6}{5}} as the division of square roots \frac{\sqrt{6}}{\sqrt{5}}.

$$\frac{\frac{2\sqrt{3}}{3}\times \frac{\sqrt{6}}{\sqrt{5}}\sqrt{\frac{1\times 7+1}{7}}}{\sqrt{\frac{1\times 10+1}{10}}}$$

Rationalize the denominator of \frac{\sqrt{6}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\frac{2\sqrt{3}}{3}\times \frac{\sqrt{6}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\sqrt{\frac{1\times 7+1}{7}}}{\sqrt{\frac{1\times 10+1}{10}}}$$

The square of \sqrt{5} is 5.

$$\frac{\frac{2\sqrt{3}}{3}\times \frac{\sqrt{6}\sqrt{5}}{5}\sqrt{\frac{1\times 7+1}{7}}}{\sqrt{\frac{1\times 10+1}{10}}}$$

To multiply \sqrt{6} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\frac{2\sqrt{3}}{3}\times \frac{\sqrt{30}}{5}\sqrt{\frac{1\times 7+1}{7}}}{\sqrt{\frac{1\times 10+1}{10}}}$$

Multiply 1 and 7 to get 7.

$$\frac{\frac{2\sqrt{3}}{3}\times \frac{\sqrt{30}}{5}\sqrt{\frac{7+1}{7}}}{\sqrt{\frac{1\times 10+1}{10}}}$$

Add 7 and 1 to get 8.

$$\frac{\frac{2\sqrt{3}}{3}\times \frac{\sqrt{30}}{5}\sqrt{\frac{8}{7}}}{\sqrt{\frac{1\times 10+1}{10}}}$$

Rewrite the square root of the division \sqrt{\frac{8}{7}} as the division of square roots \frac{\sqrt{8}}{\sqrt{7}}.

$$\frac{\frac{2\sqrt{3}}{3}\times \frac{\sqrt{30}}{5}\times \frac{\sqrt{8}}{\sqrt{7}}}{\sqrt{\frac{1\times 10+1}{10}}}$$

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}.

$$\frac{\frac{2\sqrt{3}}{3}\times \frac{\sqrt{30}}{5}\times \frac{2\sqrt{2}}{\sqrt{7}}}{\sqrt{\frac{1\times 10+1}{10}}}$$

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

$$\frac{\frac{2\sqrt{3}}{3}\times \frac{\sqrt{30}}{5}\times \frac{2\sqrt{2}\sqrt{7}}{\left(\sqrt{7}\right)^{2}}}{\sqrt{\frac{1\times 10+1}{10}}}$$

The square of \sqrt{7} is 7.

$$\frac{\frac{2\sqrt{3}}{3}\times \frac{\sqrt{30}}{5}\times \frac{2\sqrt{2}\sqrt{7}}{7}}{\sqrt{\frac{1\times 10+1}{10}}}$$

To multiply \sqrt{2} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{\frac{2\sqrt{3}}{3}\times \frac{\sqrt{30}}{5}\times \frac{2\sqrt{14}}{7}}{\sqrt{\frac{1\times 10+1}{10}}}$$

Multiply \frac{2\sqrt{3}}{3} times \frac{\sqrt{30}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{2\sqrt{3}\sqrt{30}}{3\times 5}\times \frac{2\sqrt{14}}{7}}{\sqrt{\frac{1\times 10+1}{10}}}$$

Multiply \frac{2\sqrt{3}\sqrt{30}}{3\times 5} times \frac{2\sqrt{14}}{7} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{2\sqrt{3}\sqrt{30}\times 2\sqrt{14}}{3\times 5\times 7}}{\sqrt{\frac{1\times 10+1}{10}}}$$

Multiply 1 and 10 to get 10.

$$\frac{\frac{2\sqrt{3}\sqrt{30}\times 2\sqrt{14}}{3\times 5\times 7}}{\sqrt{\frac{10+1}{10}}}$$

Add 10 and 1 to get 11.

$$\frac{\frac{2\sqrt{3}\sqrt{30}\times 2\sqrt{14}}{3\times 5\times 7}}{\sqrt{\frac{11}{10}}}$$

Rewrite the square root of the division \sqrt{\frac{11}{10}} as the division of square roots \frac{\sqrt{11}}{\sqrt{10}}.

$$\frac{\frac{2\sqrt{3}\sqrt{30}\times 2\sqrt{14}}{3\times 5\times 7}}{\frac{\sqrt{11}}{\sqrt{10}}}$$

Rationalize the denominator of \frac{\sqrt{11}}{\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{\frac{2\sqrt{3}\sqrt{30}\times 2\sqrt{14}}{3\times 5\times 7}}{\frac{\sqrt{11}\sqrt{10}}{\left(\sqrt{10}\right)^{2}}}$$

The square of \sqrt{10} is 10.

$$\frac{\frac{2\sqrt{3}\sqrt{30}\times 2\sqrt{14}}{3\times 5\times 7}}{\frac{\sqrt{11}\sqrt{10}}{10}}$$

To multiply \sqrt{11} and \sqrt{10}, multiply the numbers under the square root.

$$\frac{\frac{2\sqrt{3}\sqrt{30}\times 2\sqrt{14}}{3\times 5\times 7}}{\frac{\sqrt{110}}{10}}$$

Divide \frac{2\sqrt{3}\sqrt{30}\times 2\sqrt{14}}{3\times 5\times 7} by \frac{\sqrt{110}}{10} by multiplying \frac{2\sqrt{3}\sqrt{30}\times 2\sqrt{14}}{3\times 5\times 7} by the reciprocal of \frac{\sqrt{110}}{10}.

$$\frac{2\sqrt{3}\sqrt{30}\times 2\sqrt{14}\times 10}{3\times 5\times 7\sqrt{110}}$$

Cancel out 5 in both numerator and denominator.

$$\frac{2\times 2\times 2\sqrt{3}\sqrt{14}\sqrt{30}}{3\times 7\sqrt{110}}$$

Rationalize the denominator of \frac{2\times 2\times 2\sqrt{3}\sqrt{14}\sqrt{30}}{3\times 7\sqrt{110}} by multiplying numerator and denominator by \sqrt{110}.

$$\frac{2\times 2\times 2\sqrt{3}\sqrt{14}\sqrt{30}\sqrt{110}}{3\times 7\left(\sqrt{110}\right)^{2}}$$

The square of \sqrt{110} is 110.

$$\frac{2\times 2\times 2\sqrt{3}\sqrt{14}\sqrt{30}\sqrt{110}}{3\times 7\times 110}$$

Factor 30=3\times 10. Rewrite the square root of the product \sqrt{3\times 10} as the product of square roots \sqrt{3}\sqrt{10}.

$$\frac{2\times 2\times 2\sqrt{3}\sqrt{14}\sqrt{3}\sqrt{10}\sqrt{110}}{3\times 7\times 110}$$

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

$$\frac{2\times 2\times 2\times 3\sqrt{14}\sqrt{10}\sqrt{110}}{3\times 7\times 110}$$

Multiply 2 and 2 to get 4.

$$\frac{4\times 2\times 3\sqrt{14}\sqrt{10}\sqrt{110}}{3\times 7\times 110}$$

Multiply 4 and 2 to get 8.

$$\frac{8\times 3\sqrt{14}\sqrt{10}\sqrt{110}}{3\times 7\times 110}$$

Multiply 8 and 3 to get 24.

$$\frac{24\sqrt{14}\sqrt{10}\sqrt{110}}{3\times 7\times 110}$$

To multiply \sqrt{14} and \sqrt{10}, multiply the numbers under the square root.

$$\frac{24\sqrt{140}\sqrt{110}}{3\times 7\times 110}$$

Factor 140=2^{2}\times 35. Rewrite the square root of the product \sqrt{2^{2}\times 35} as the product of square roots \sqrt{2^{2}}\sqrt{35}. Take the square root of 2^{2}.

$$\frac{24\times 2\sqrt{35}\sqrt{110}}{3\times 7\times 110}$$

Multiply 24 and 2 to get 48.

$$\frac{48\sqrt{35}\sqrt{110}}{3\times 7\times 110}$$

To multiply \sqrt{35} and \sqrt{110}, multiply the numbers under the square root.

$$\frac{48\sqrt{3850}}{3\times 7\times 110}$$

Multiply 3 and 7 to get 21.

$$\frac{48\sqrt{3850}}{21\times 110}$$

Multiply 21 and 110 to get 2310.

$$\frac{48\sqrt{3850}}{2310}$$

Factor 3850=5^{2}\times 154. Rewrite the square root of the product \sqrt{5^{2}\times 154} as the product of square roots \sqrt{5^{2}}\sqrt{154}. Take the square root of 5^{2}.

$$\frac{48\times 5\sqrt{154}}{2310}$$

Multiply 48 and 5 to get 240.

$$\frac{240\sqrt{154}}{2310}$$

Divide 240\sqrt{154} by 2310 to get \frac{8}{77}\sqrt{154}.

$$\frac{8}{77}\sqrt{154}$$