Solve sqrt{7/11}left(sqrt{11/7}+sqrt{1/11}right)

$$\sqrt{ \frac{ 7 }{ 11 } } \left( \sqrt{ \frac{ 11 }{ 7 } } + \sqrt{ \frac{ 1 }{ 11 } } \right)$$

Evaluate

$$\frac{\sqrt{7}}{11}+1\approx 1.240522846$$

Short Solution Steps

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

$$\frac{\sqrt{7}}{\sqrt{11}}\left(\sqrt{\frac{11}{7}}+\sqrt{\frac{1}{11}}\right)$$

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

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

The square of \sqrt{11} is 11.

$$\frac{\sqrt{7}\sqrt{11}}{11}\left(\sqrt{\frac{11}{7}}+\sqrt{\frac{1}{11}}\right)$$

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

$$\frac{\sqrt{77}}{11}\left(\sqrt{\frac{11}{7}}+\sqrt{\frac{1}{11}}\right)$$

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

$$\frac{\sqrt{77}}{11}\left(\frac{\sqrt{11}}{\sqrt{7}}+\sqrt{\frac{1}{11}}\right)$$

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

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

The square of \sqrt{7} is 7.

$$\frac{\sqrt{77}}{11}\left(\frac{\sqrt{11}\sqrt{7}}{7}+\sqrt{\frac{1}{11}}\right)$$

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

$$\frac{\sqrt{77}}{11}\left(\frac{\sqrt{77}}{7}+\sqrt{\frac{1}{11}}\right)$$

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

$$\frac{\sqrt{77}}{11}\left(\frac{\sqrt{77}}{7}+\frac{\sqrt{1}}{\sqrt{11}}\right)$$

Calculate the square root of 1 and get 1.

$$\frac{\sqrt{77}}{11}\left(\frac{\sqrt{77}}{7}+\frac{1}{\sqrt{11}}\right)$$

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

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

The square of \sqrt{11} is 11.

$$\frac{\sqrt{77}}{11}\left(\frac{\sqrt{77}}{7}+\frac{\sqrt{11}}{11}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 7 and 11 is 77. Multiply \frac{\sqrt{77}}{7} times \frac{11}{11}. Multiply \frac{\sqrt{11}}{11} times \frac{7}{7}.

$$\frac{\sqrt{77}}{11}\left(\frac{11\sqrt{77}}{77}+\frac{7\sqrt{11}}{77}\right)$$

Since \frac{11\sqrt{77}}{77} and \frac{7\sqrt{11}}{77} have the same denominator, add them by adding their numerators.

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

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

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

Multiply 11 and 77 to get 847.

$$\frac{\sqrt{77}\left(11\sqrt{77}+7\sqrt{11}\right)}{847}$$

Use the distributive property to multiply \sqrt{77} by 11\sqrt{77}+7\sqrt{11}.

$$\frac{11\left(\sqrt{77}\right)^{2}+7\sqrt{77}\sqrt{11}}{847}$$

The square of \sqrt{77} is 77.

$$\frac{11\times 77+7\sqrt{77}\sqrt{11}}{847}$$

Multiply 11 and 77 to get 847.

$$\frac{847+7\sqrt{77}\sqrt{11}}{847}$$

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}.

$$\frac{847+7\sqrt{11}\sqrt{7}\sqrt{11}}{847}$$

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

$$\frac{847+7\times 11\sqrt{7}}{847}$$

Multiply 7 and 11 to get 77.

$$\frac{847+77\sqrt{7}}{847}$$