Solve 4/1+sqrt{5}

$$\frac{ 4 }{ 1+ \sqrt{ 5 } }$$

Evaluate

$$\sqrt{5}-1\approx 1.236067977$$

Solution Steps

Rationalize the denominator of \frac{4}{1+\sqrt{5}} by multiplying numerator and denominator by 1-\sqrt{5}.

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

Consider \left(1+\sqrt{5}\right)\left(1-\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}.

$$\frac{4\left(1-\sqrt{5}\right)}{1^{2}-\left(\sqrt{5}\right)^{2}}$$

Square 1. Square \sqrt{5}.

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

Subtract 5 from 1 to get -4.

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

Cancel out -4 and -4.

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

To find the opposite of 1-\sqrt{5}, find the opposite of each term.

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

The opposite of -\sqrt{5} is \sqrt{5}.

$$-1+\sqrt{5}$$