Solve sqrt{7/11}*(sqrt{11/7}divsqrt{1/11}

$$\sqrt{ \frac{ 7 }{ 11 } } \times ( \sqrt{ \frac{ 11 }{ 7 } } \div \sqrt{ \frac{ 1 }{ 11 } }$$

Evaluate

$$\sqrt{11}\approx 3.31662479$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{7}{11}} as the division of square roots \frac{\sqrt{7}}{\sqrt{11}}.

$$\frac{\sqrt{7}}{\sqrt{11}}\times \frac{\sqrt{\frac{11}{7}}}{\sqrt{\frac{1}{11}}}$$

Rationalize the denominator of \frac{\sqrt{7}}{\sqrt{11}} by multiplying numerator and denominator by \sqrt{11}.

$$\frac{\sqrt{7}\sqrt{11}}{\left(\sqrt{11}\right)^{2}}\times \frac{\sqrt{\frac{11}{7}}}{\sqrt{\frac{1}{11}}}$$

The square of \sqrt{11} is 11.

$$\frac{\sqrt{7}\sqrt{11}}{11}\times \frac{\sqrt{\frac{11}{7}}}{\sqrt{\frac{1}{11}}}$$

To multiply \sqrt{7} and \sqrt{11}, multiply the numbers under the square root.

$$\frac{\sqrt{77}}{11}\times \frac{\sqrt{\frac{11}{7}}}{\sqrt{\frac{1}{11}}}$$

Rewrite the square root of the division \sqrt{\frac{11}{7}} as the division of square roots \frac{\sqrt{11}}{\sqrt{7}}.

$$\frac{\sqrt{77}}{11}\times \frac{\frac{\sqrt{11}}{\sqrt{7}}}{\sqrt{\frac{1}{11}}}$$

Rationalize the denominator of \frac{\sqrt{11}}{\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{\sqrt{77}}{11}\times \frac{\frac{\sqrt{11}\sqrt{7}}{\left(\sqrt{7}\right)^{2}}}{\sqrt{\frac{1}{11}}}$$

The square of \sqrt{7} is 7.

$$\frac{\sqrt{77}}{11}\times \frac{\frac{\sqrt{11}\sqrt{7}}{7}}{\sqrt{\frac{1}{11}}}$$

To multiply \sqrt{11} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{\sqrt{77}}{11}\times \frac{\frac{\sqrt{77}}{7}}{\sqrt{\frac{1}{11}}}$$

Rewrite the square root of the division \sqrt{\frac{1}{11}} as the division of square roots \frac{\sqrt{1}}{\sqrt{11}}.

$$\frac{\sqrt{77}}{11}\times \frac{\frac{\sqrt{77}}{7}}{\frac{\sqrt{1}}{\sqrt{11}}}$$

Calculate the square root of 1 and get 1.

$$\frac{\sqrt{77}}{11}\times \frac{\frac{\sqrt{77}}{7}}{\frac{1}{\sqrt{11}}}$$

Rationalize the denominator of \frac{1}{\sqrt{11}} by multiplying numerator and denominator by \sqrt{11}.

$$\frac{\sqrt{77}}{11}\times \frac{\frac{\sqrt{77}}{7}}{\frac{\sqrt{11}}{\left(\sqrt{11}\right)^{2}}}$$

The square of \sqrt{11} is 11.

$$\frac{\sqrt{77}}{11}\times \frac{\frac{\sqrt{77}}{7}}{\frac{\sqrt{11}}{11}}$$

Divide \frac{\sqrt{77}}{7} by \frac{\sqrt{11}}{11} by multiplying \frac{\sqrt{77}}{7} by the reciprocal of \frac{\sqrt{11}}{11}.

$$\frac{\sqrt{77}}{11}\times \frac{\sqrt{77}\times 11}{7\sqrt{11}}$$

Rationalize the denominator of \frac{\sqrt{77}\times 11}{7\sqrt{11}} by multiplying numerator and denominator by \sqrt{11}.

$$\frac{\sqrt{77}}{11}\times \frac{\sqrt{77}\times 11\sqrt{11}}{7\left(\sqrt{11}\right)^{2}}$$

The square of \sqrt{11} is 11.

$$\frac{\sqrt{77}}{11}\times \frac{\sqrt{77}\times 11\sqrt{11}}{7\times 11}$$

Factor 77=11\times 7. Rewrite the square root of the product \sqrt{11\times 7} as the product of square roots \sqrt{11}\sqrt{7}.

$$\frac{\sqrt{77}}{11}\times \frac{\sqrt{11}\sqrt{7}\times 11\sqrt{11}}{7\times 11}$$

Multiply \sqrt{11} and \sqrt{11} to get 11.

$$\frac{\sqrt{77}}{11}\times \frac{11\times 11\sqrt{7}}{7\times 11}$$

Multiply 7 and 11 to get 77.

$$\frac{\sqrt{77}}{11}\times \frac{11\times 11\sqrt{7}}{77}$$

Multiply 11 and 11 to get 121.

$$\frac{\sqrt{77}}{11}\times \frac{121\sqrt{7}}{77}$$

Divide 121\sqrt{7} by 77 to get \frac{11}{7}\sqrt{7}.

$$\frac{\sqrt{77}}{11}\times \frac{11}{7}\sqrt{7}$$

Multiply \frac{\sqrt{77}}{11} times \frac{11}{7} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{77}\times 11}{11\times 7}\sqrt{7}$$

Cancel out 11 in both numerator and denominator.

$$\frac{\sqrt{77}}{7}\sqrt{7}$$

Express \frac{\sqrt{77}}{7}\sqrt{7} as a single fraction.

$$\frac{\sqrt{77}\sqrt{7}}{7}$$

Factor 77=7\times 11. Rewrite the square root of the product \sqrt{7\times 11} as the product of square roots \sqrt{7}\sqrt{11}.

$$\frac{\sqrt{7}\sqrt{11}\sqrt{7}}{7}$$

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

$$\frac{7\sqrt{11}}{7}$$

Cancel out 7 and 7.

$$\sqrt{11}$$