Rationalizing a Denominator Containing Two Terms

Write $\frac{4}{1 + \sqrt{5}}$ in simplest form.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Begin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate of $1 + \sqrt{5}$ is $1 - \sqrt{5} .$ Then multiply the fraction by $\frac{1 - \sqrt{5}}{1 - \sqrt{5}} .$

$\begin{matrix} \frac{4}{1 + \sqrt{5}} \cdot \frac{1 - \sqrt{5}}{1 - \sqrt{5}} \\ \frac{4 - 4 \sqrt{5}}{- 4} & Use the distributive property . \\ \sqrt{5} - 1 & Simplify . \end{matrix}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Write $\frac{7}{2 + \sqrt{3}}$ in simplest form.

How did it go?