Solve (frac{sqrt{7}}{sqrt{7}-sqrt{5}}+frac{sqrt{5}}{sqrt{7}-sqrt{5}})+6/sqrt{5}

$$( \frac { \sqrt { 7 } } { \sqrt { 7 } - \sqrt { 5 } } + \frac { \sqrt { 5 } } { \sqrt { 7 } - \sqrt { 5 } } ) + \frac { 6 } { \sqrt { 5 } }$$

Evaluate

$$\frac{6\sqrt{5}}{5}+\sqrt{35}+6\approx 14.599361356$$

Short Solution Steps

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

$$\frac{\sqrt{7}\left(\sqrt{7}+\sqrt{5}\right)}{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}+\frac{\sqrt{5}}{\sqrt{7}-\sqrt{5}}+\frac{6}{\sqrt{5}}$$

Consider \left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{7}. Square \sqrt{5}.

$$\frac{\sqrt{7}\left(\sqrt{7}+\sqrt{5}\right)}{7-5}+\frac{\sqrt{5}}{\sqrt{7}-\sqrt{5}}+\frac{6}{\sqrt{5}}$$

Subtract 5 from 7 to get 2.

$$\frac{\sqrt{7}\left(\sqrt{7}+\sqrt{5}\right)}{2}+\frac{\sqrt{5}}{\sqrt{7}-\sqrt{5}}+\frac{6}{\sqrt{5}}$$

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

$$\frac{\sqrt{7}\left(\sqrt{7}+\sqrt{5}\right)}{2}+\frac{\sqrt{5}\left(\sqrt{7}+\sqrt{5}\right)}{\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)}+\frac{6}{\sqrt{5}}$$

Consider \left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{7}. Square \sqrt{5}.

$$\frac{\sqrt{7}\left(\sqrt{7}+\sqrt{5}\right)}{2}+\frac{\sqrt{5}\left(\sqrt{7}+\sqrt{5}\right)}{7-5}+\frac{6}{\sqrt{5}}$$

Subtract 5 from 7 to get 2.

$$\frac{\sqrt{7}\left(\sqrt{7}+\sqrt{5}\right)}{2}+\frac{\sqrt{5}\left(\sqrt{7}+\sqrt{5}\right)}{2}+\frac{6}{\sqrt{5}}$$

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

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

The square of \sqrt{5} is 5.

$$\frac{\sqrt{7}\left(\sqrt{7}+\sqrt{5}\right)}{2}+\frac{\sqrt{5}\left(\sqrt{7}+\sqrt{5}\right)}{2}+\frac{6\sqrt{5}}{5}$$

Since \frac{\sqrt{7}\left(\sqrt{7}+\sqrt{5}\right)}{2} and \frac{\sqrt{5}\left(\sqrt{7}+\sqrt{5}\right)}{2} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{7}\left(\sqrt{7}+\sqrt{5}\right)+\sqrt{5}\left(\sqrt{7}+\sqrt{5}\right)}{2}+\frac{6\sqrt{5}}{5}$$

Do the multiplications in \sqrt{7}\left(\sqrt{7}+\sqrt{5}\right)+\sqrt{5}\left(\sqrt{7}+\sqrt{5}\right).

$$\frac{7+\sqrt{35}+\sqrt{35}+5}{2}+\frac{6\sqrt{5}}{5}$$

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

$$\frac{12+2\sqrt{35}}{2}+\frac{6\sqrt{5}}{5}$$

Divide each term of 12+2\sqrt{35} by 2 to get 6+\sqrt{35}.

$$6+\sqrt{35}+\frac{6\sqrt{5}}{5}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 6+\sqrt{35} times \frac{5}{5}.

$$\frac{5\left(6+\sqrt{35}\right)}{5}+\frac{6\sqrt{5}}{5}$$

Since \frac{5\left(6+\sqrt{35}\right)}{5} and \frac{6\sqrt{5}}{5} have the same denominator, add them by adding their numerators.

$$\frac{5\left(6+\sqrt{35}\right)+6\sqrt{5}}{5}$$

Do the multiplications in 5\left(6+\sqrt{35}\right)+6\sqrt{5}.

$$\frac{30+5\sqrt{35}+6\sqrt{5}}{5}$$