Question

In the following exercises, simplify by rationalizing the denominator

(a)$\frac{3}{3+\sqrt{11}}$ (b)$\frac{8}{1-\sqrt{5}}$

Short answer

(a) $-\frac{3\left(3-\sqrt{11}\right)}{2}$

(b)$-2\left(1+\sqrt{5}\right)$

Step-by-step solution

Step 1. Given Information

(a)$\frac{3}{3+\sqrt{11}}$(b)$\frac{8}{1-\sqrt{5}}$

Part (a) of Step 1. Rationalize

$$\begin{gathered} \frac{3}{3+\sqrt{11}} \\ \frac{3}{3+\sqrt{11}}\times \frac{3-\sqrt{11}}{3-\sqrt{11}} \\ \frac{3\left(3-\sqrt{11}\right)}{9-11} \\ \frac{3\left(3-\sqrt{11}\right)}{-2} \end{gathered}$$

Part (b) of Step 1. Rationalize

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