Q. 271 - Intermediate Algebra

Question

In the following exercises, simplify.

$\frac{8}{1-\sqrt{5}}$

Step-by-Step Solution

Answer

The simplified form is $-2\left(1+\sqrt{5}\right)$.

Step 1. Given information.

The given expression is,

$\frac{8}{1-\sqrt{5}}$.

Step 2 Simplifying.

Now, to rationalize we will multiply the numerator and denominator by the conjugate of the denominator and simplify.

$$\begin{gathered} \frac{8}{1-\sqrt{5}}=\frac{8\left(1+\sqrt{5}\right)}{\left(1-\sqrt{5}\right)\left(1+\sqrt{5}\right)} \\ =\frac{8\left(1+\sqrt{5}\right)}{1^2-{\left(\sqrt{5}\right)}^2} \\ =\frac{8\left(1+\sqrt{5}\right)}{1-5} \\ =\frac{8\left(1+\sqrt{5}\right)}{-4} \\ =-2\left(1+\sqrt{5}\right) \end{gathered}$$