Question

In the following exercises, simplify by rationalizing the denominator

(a)$\frac{6}{6+\sqrt{5}}$ (b)$\frac{5}{4-\sqrt{11}}$

Short answer

(a) $\frac{6\left(6-\sqrt{5}\right)}{31}$

(b)$4+\sqrt{11}$

Step-by-step solution

Step 1. Given Information

(a)$\frac{6}{6+\sqrt{5}}$ (b)$\frac{5}{4-\sqrt{11}}$

Part (a) of Step 1. Rationalize

$$\begin{gathered} \frac{6}{6+\sqrt{5}} \\ \frac{6}{6+\sqrt{5}}\times \frac{6-\sqrt{5}}{6-\sqrt{5}} \\ \frac{6\left(6-\sqrt{5}\right)}{\left(6+\sqrt{5}\right)\left(6-\sqrt{5}\right)} \\ \frac{6\left(6-\sqrt{5}\right)}{31} \end{gathered}$$

Part (b) of Step 1. Rationalize

$$\begin{gathered} \frac{5}{4-\sqrt{11}} \\ \frac{5}{4-\sqrt{11}}\times \frac{4+\sqrt{11}}{4+\sqrt{11}} \\ \frac{5\left(4+\sqrt{11}\right)}{\left(4-\sqrt{11}\right)\left(4+\sqrt{11}\right)} \\ \frac{5\left(4+\sqrt{11}\right)}{5} \\ 4+\sqrt{11} \end{gathered}$$