Solve frac{sqrt{13}+sqrt{5}}{sqrt{13}-sqrt{5}}

$$\frac { \sqrt { 13 } + \sqrt { 5 } } { \sqrt { 13 } - \sqrt { 5 } }$$

Evaluate

$$\frac{\sqrt{65}+9}{4}\approx 4.265564437$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{13}+\sqrt{5}}{\sqrt{13}-\sqrt{5}} by multiplying numerator and denominator by \sqrt{13}+\sqrt{5}.

$$\frac{\left(\sqrt{13}+\sqrt{5}\right)\left(\sqrt{13}+\sqrt{5}\right)}{\left(\sqrt{13}-\sqrt{5}\right)\left(\sqrt{13}+\sqrt{5}\right)}$$

Consider \left(\sqrt{13}-\sqrt{5}\right)\left(\sqrt{13}+\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}.

$$\frac{\left(\sqrt{13}+\sqrt{5}\right)\left(\sqrt{13}+\sqrt{5}\right)}{\left(\sqrt{13}\right)^{2}-\left(\sqrt{5}\right)^{2}}$$

Square \sqrt{13}. Square \sqrt{5}.

$$\frac{\left(\sqrt{13}+\sqrt{5}\right)\left(\sqrt{13}+\sqrt{5}\right)}{13-5}$$

Subtract 5 from 13 to get 8.

$$\frac{\left(\sqrt{13}+\sqrt{5}\right)\left(\sqrt{13}+\sqrt{5}\right)}{8}$$

Multiply \sqrt{13}+\sqrt{5} and \sqrt{13}+\sqrt{5} to get \left(\sqrt{13}+\sqrt{5}\right)^{2}.

$$\frac{\left(\sqrt{13}+\sqrt{5}\right)^{2}}{8}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{13}+\sqrt{5}\right)^{2}.

$$\frac{\left(\sqrt{13}\right)^{2}+2\sqrt{13}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{8}$$

The square of \sqrt{13} is 13.

$$\frac{13+2\sqrt{13}\sqrt{5}+\left(\sqrt{5}\right)^{2}}{8}$$

To multiply \sqrt{13} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{13+2\sqrt{65}+\left(\sqrt{5}\right)^{2}}{8}$$

The square of \sqrt{5} is 5.

$$\frac{13+2\sqrt{65}+5}{8}$$

Add 13 and 5 to get 18.

$$\frac{18+2\sqrt{65}}{8}$$