Solve sqrt{frac{7/5}+sqrt{5/7}}{sqrt{7/5}-sqrt{5/7}}

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

Evaluate

$$6$$

Short Solution Steps

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

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

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

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

The square of \sqrt{5} is 5.

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

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

$$\frac{\frac{\sqrt{35}}{5}+\sqrt{\frac{5}{7}}}{\sqrt{\frac{7}{5}}-\sqrt{\frac{5}{7}}}$$

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

$$\frac{\frac{\sqrt{35}}{5}+\frac{\sqrt{5}}{\sqrt{7}}}{\sqrt{\frac{7}{5}}-\sqrt{\frac{5}{7}}}$$

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

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

The square of \sqrt{7} is 7.

$$\frac{\frac{\sqrt{35}}{5}+\frac{\sqrt{5}\sqrt{7}}{7}}{\sqrt{\frac{7}{5}}-\sqrt{\frac{5}{7}}}$$

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

$$\frac{\frac{\sqrt{35}}{5}+\frac{\sqrt{35}}{7}}{\sqrt{\frac{7}{5}}-\sqrt{\frac{5}{7}}}$$

Combine \frac{\sqrt{35}}{5} and \frac{\sqrt{35}}{7} to get \frac{12}{35}\sqrt{35}.

$$\frac{\frac{12}{35}\sqrt{35}}{\sqrt{\frac{7}{5}}-\sqrt{\frac{5}{7}}}$$

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

$$\frac{\frac{12}{35}\sqrt{35}}{\frac{\sqrt{7}}{\sqrt{5}}-\sqrt{\frac{5}{7}}}$$

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

$$\frac{\frac{12}{35}\sqrt{35}}{\frac{\sqrt{7}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}-\sqrt{\frac{5}{7}}}$$

The square of \sqrt{5} is 5.

$$\frac{\frac{12}{35}\sqrt{35}}{\frac{\sqrt{7}\sqrt{5}}{5}-\sqrt{\frac{5}{7}}}$$

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

$$\frac{\frac{12}{35}\sqrt{35}}{\frac{\sqrt{35}}{5}-\sqrt{\frac{5}{7}}}$$

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

$$\frac{\frac{12}{35}\sqrt{35}}{\frac{\sqrt{35}}{5}-\frac{\sqrt{5}}{\sqrt{7}}}$$

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

$$\frac{\frac{12}{35}\sqrt{35}}{\frac{\sqrt{35}}{5}-\frac{\sqrt{5}\sqrt{7}}{\left(\sqrt{7}\right)^{2}}}$$

The square of \sqrt{7} is 7.

$$\frac{\frac{12}{35}\sqrt{35}}{\frac{\sqrt{35}}{5}-\frac{\sqrt{5}\sqrt{7}}{7}}$$

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

$$\frac{\frac{12}{35}\sqrt{35}}{\frac{\sqrt{35}}{5}-\frac{\sqrt{35}}{7}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 7 is 35. Multiply \frac{\sqrt{35}}{5} times \frac{7}{7}. Multiply \frac{\sqrt{35}}{7} times \frac{5}{5}.

$$\frac{\frac{12}{35}\sqrt{35}}{\frac{7\sqrt{35}}{35}-\frac{5\sqrt{35}}{35}}$$

Since \frac{7\sqrt{35}}{35} and \frac{5\sqrt{35}}{35} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{12}{35}\sqrt{35}}{\frac{7\sqrt{35}-5\sqrt{35}}{35}}$$

Do the calculations in 7\sqrt{35}-5\sqrt{35}.

$$\frac{\frac{12}{35}\sqrt{35}}{\frac{2\sqrt{35}}{35}}$$

Divide \frac{12}{35}\sqrt{35} by \frac{2\sqrt{35}}{35} by multiplying \frac{12}{35}\sqrt{35} by the reciprocal of \frac{2\sqrt{35}}{35}.

$$\frac{\frac{12}{35}\sqrt{35}\times 35}{2\sqrt{35}}$$

Cancel out \sqrt{35} in both numerator and denominator.

$$\frac{\frac{12}{35}\times 35}{2}$$

Cancel out 35 and 35.

$$\frac{12}{2}$$

Divide 12 by 2 to get 6.

$$6$$

Factor

$$2\times 3$$