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

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

Evaluate

$$\frac{\sqrt{5}-1}{2}\approx 0.618033989$$

Solution Steps

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

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

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}.

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

Square 3. Square \sqrt{5}.

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

Subtract 5 from 9 to get 4.

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

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

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

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

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

The square of \sqrt{5} is 5.

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

Subtract 5 from 3 to get -2.

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