Solve frac{sqrt{13}}{13+sqrt{13}}

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

Evaluate

$$\frac{\sqrt{13}-1}{12}\approx 0.217129273$$

Solution Steps

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

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

Consider \left(13+\sqrt{13}\right)\left(13-\sqrt{13}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square 13. Square \sqrt{13}.

$$\frac{\sqrt{13}\left(13-\sqrt{13}\right)}{169-13}$$

Subtract 13 from 169 to get 156.

$$\frac{\sqrt{13}\left(13-\sqrt{13}\right)}{156}$$

Use the distributive property to multiply \sqrt{13} by 13-\sqrt{13}.

$$\frac{13\sqrt{13}-\left(\sqrt{13}\right)^{2}}{156}$$

The square of \sqrt{13} is 13.

$$\frac{13\sqrt{13}-13}{156}$$