Solve (frac{sqrt{7}}{sqrt{7}-sqrt{5}}+frac{sqrt{5}}{sqrt{7}+sqrt{5}})div6/sqrt{5}

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

Evaluate

$$\frac{\sqrt{5}\left(\sqrt{35}+1\right)}{6}\approx 2.577470755$$

Short Solution Steps

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

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

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}.

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

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

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

Subtract 5 from 7 to get 2.

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

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

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

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}.

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

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

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

Subtract 5 from 7 to get 2.

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

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.

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

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

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

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

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

Divide each term of 2+2\sqrt{35} by 2 to get 1+\sqrt{35}.

$$\frac{1+\sqrt{35}}{\frac{6}{\sqrt{5}}}$$

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

$$\frac{1+\sqrt{35}}{\frac{6\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}$$

The square of \sqrt{5} is 5.

$$\frac{1+\sqrt{35}}{\frac{6\sqrt{5}}{5}}$$

Divide 1+\sqrt{35} by \frac{6\sqrt{5}}{5} by multiplying 1+\sqrt{35} by the reciprocal of \frac{6\sqrt{5}}{5}.

$$\frac{\left(1+\sqrt{35}\right)\times 5}{6\sqrt{5}}$$

Rationalize the denominator of \frac{\left(1+\sqrt{35}\right)\times 5}{6\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\left(1+\sqrt{35}\right)\times 5\sqrt{5}}{6\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(1+\sqrt{35}\right)\times 5\sqrt{5}}{6\times 5}$$

Multiply 6 and 5 to get 30.

$$\frac{\left(1+\sqrt{35}\right)\times 5\sqrt{5}}{30}$$

Divide \left(1+\sqrt{35}\right)\times 5\sqrt{5} by 30 to get \left(1+\sqrt{35}\right)\times \frac{1}{6}\sqrt{5}.

$$\left(1+\sqrt{35}\right)\times \frac{1}{6}\sqrt{5}$$

Use the distributive property to multiply 1+\sqrt{35} by \frac{1}{6}.

$$\left(\frac{1}{6}+\sqrt{35}\times \frac{1}{6}\right)\sqrt{5}$$

Use the distributive property to multiply \frac{1}{6}+\sqrt{35}\times \frac{1}{6} by \sqrt{5}.

$$\frac{1}{6}\sqrt{5}+\sqrt{35}\times \frac{1}{6}\sqrt{5}$$

Factor 35=5\times 7. Rewrite the square root of the product \sqrt{5\times 7} as the product of square roots \sqrt{5}\sqrt{7}.

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

Multiply \sqrt{5} and \sqrt{5} to get 5.

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

Multiply 5 and \frac{1}{6} to get \frac{5}{6}.

$$\frac{1}{6}\sqrt{5}+\frac{5}{6}\sqrt{7}$$