Solve frac{sqrt{5}-1}{3-sqrt{5}}

$$\frac { \sqrt { 5 } - 1 } { 3 - \sqrt { 5 } }$$

Evaluate

$$\frac{\sqrt{5}+1}{2}\approx 1.618033989$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{5}-1}{3-\sqrt{5}} by multiplying numerator and denominator by 3+\sqrt{5}.

$$\frac{\left(\sqrt{5}-1\right)\left(3+\sqrt{5}\right)}{\left(3-\sqrt{5}\right)\left(3+\sqrt{5}\right)}$$

Consider \left(3-\sqrt{5}\right)\left(3+\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}.

$$\frac{\left(\sqrt{5}-1\right)\left(3+\sqrt{5}\right)}{3^{2}-\left(\sqrt{5}\right)^{2}}$$

Square 3. Square \sqrt{5}.

$$\frac{\left(\sqrt{5}-1\right)\left(3+\sqrt{5}\right)}{9-5}$$

Subtract 5 from 9 to get 4.

$$\frac{\left(\sqrt{5}-1\right)\left(3+\sqrt{5}\right)}{4}$$

Apply the distributive property by multiplying each term of \sqrt{5}-1 by each term of 3+\sqrt{5}.

$$\frac{3\sqrt{5}+\left(\sqrt{5}\right)^{2}-3-\sqrt{5}}{4}$$

The square of \sqrt{5} is 5.

$$\frac{3\sqrt{5}+5-3-\sqrt{5}}{4}$$

Subtract 3 from 5 to get 2.

$$\frac{3\sqrt{5}+2-\sqrt{5}}{4}$$

Combine 3\sqrt{5} and -\sqrt{5} to get 2\sqrt{5}.

$$\frac{2\sqrt{5}+2}{4}$$