Question

In the following exercises, simplify by rationalizing the denominator

(a)$\frac{4}{4+\sqrt{7}}$ (b)$\frac{7}{2-\sqrt{6}}$

Short answer

(a) $\frac{4\left(4-\sqrt{7}\right)}{9}$

(b)$-\frac{7\left(2+\sqrt{6}\right)}{2}$

Step-by-step solution

Step 1. Given Information 

(a) $\frac{4}{4+\sqrt{7}}$ (b)$\frac{7}{2-\sqrt{6}}$

Part (a) of Step 1. Rationalize

$$\begin{gathered} \frac{4}{4+\sqrt{7}} \\ \frac{4}{4+\sqrt{7}}\times \frac{4-\sqrt{7}}{4-\sqrt{7}} \\ \frac{4\left(4-\sqrt{7}\right)}{\left(4+\sqrt{7}\right)\left(4-\sqrt{7}\right)} \\ \frac{4\left(4-\sqrt{7}\right)}{9} \end{gathered}$$

Part (b) of Step 1. Rationalize

$$\begin{gathered} \frac{7}{2-\sqrt{6}} \\ \frac{7}{2-\sqrt{6}}\times \frac{2+\sqrt{6}}{2+\sqrt{6}} \\ \frac{7\left(2+\sqrt{6}\right)}{\left(2-\sqrt{6}\right)\left(2+\sqrt{6}\right)} \\ -\frac{7\left(2+\sqrt{6}\right)}{2} \end{gathered}$$