Solve 4/1-sqrt{5}*frac{1-sqrt{5}}{1-sqrt{5}}

$$\frac{ 4 }{ 1- \sqrt{ 5 } } \times \frac{ 1- \sqrt{ 5 } }{ 1- \sqrt{ 5 } }$$

Evaluate

$$-\sqrt{5}-1\approx -3.236067977$$

Solution Steps

Divide 1-\sqrt{5} by 1-\sqrt{5} to get 1.

$$\frac{4}{1-\sqrt{5}}\times 1$$

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

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

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}.

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

Square 1. Square \sqrt{5}.

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

Subtract 5 from 1 to get -4.

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

Cancel out -4 and -4.

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

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

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

Use the distributive property to multiply -1-\sqrt{5} by 1.

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

Factor

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