Q. 595 - Intermediate Algebra

Question

In the following exercises, simplify. Assume all variables are positive.

$\frac{3}{2+\sqrt{3}}$

Step-by-Step Solution

Answer

The simplified expression is $6-3\sqrt{3}$

Step 1. Given Information

We have $\frac{3}{2+\sqrt{3}}$

Step 2. Rationalize the denominator

$\frac{3}{2+\sqrt{3}}=\frac{3}{2+\sqrt{3}}\times \frac{2-\sqrt{3}}{2-\sqrt{3}}$

Step 3. Simplify the expression

$$\begin{gathered} =\frac{3\left(2-\sqrt{3}\right)}{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)} \\ =\frac{6-3\sqrt{3}}{4-3}=\frac{6-3\sqrt{3}}{1} \\ =6-3\sqrt{3} \end{gathered}$$