Solve frac{sqrt{5}}{6+sqrt{5}}

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

Evaluate

$$\frac{6\sqrt{5}-5}{31}\approx 0.271497028$$

Solution Steps

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

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

Consider \left(6+\sqrt{5}\right)\left(6-\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}.

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

Square 6. Square \sqrt{5}.

$$\frac{\sqrt{5}\left(6-\sqrt{5}\right)}{36-5}$$

Subtract 5 from 36 to get 31.

$$\frac{\sqrt{5}\left(6-\sqrt{5}\right)}{31}$$

Use the distributive property to multiply \sqrt{5} by 6-\sqrt{5}.

$$\frac{6\sqrt{5}-\left(\sqrt{5}\right)^{2}}{31}$$

The square of \sqrt{5} is 5.

$$\frac{6\sqrt{5}-5}{31}$$