# How do you rationalize the denominator and simplify (4 (sqrt3) + 2)/(2 (sqrt 3) + 1)?

Apr 19, 2018

2

#### Explanation:

1. Multiply by the conjugate (see here) of the denominator, $2 \left(\setminus \sqrt{3}\right) - 1$.
$\frac{4 \left(\sqrt{3}\right) + 2}{2 \left(\sqrt{3}\right) + 1} \times \setminus \textcolor{c r i m s o n}{\frac{2 \setminus \sqrt{3} - 1}{2 \setminus \sqrt{3} - 1}}$
2. Re-format expression.
$\frac{\left(4 \setminus \sqrt{3} + 2\right) \left(2 \setminus \sqrt{3} - 1\right)}{\left(2 \setminus \sqrt{3} + 1\right) \left(2 \setminus \sqrt{3} - 1\right)}$
3. FOIL numerator and denominator
$\frac{8 \left(3\right) - 4 \setminus \sqrt{3} + 4 \setminus \sqrt{3} - 2}{4 \left(3\right) - 2 \setminus \sqrt{3} + 2 \setminus \sqrt{3} - 1}$
4. Simplify.
$\frac{24 - 2}{12 - 1}$
$\frac{22}{11}$
5. Simplify again:
$\frac{22}{11} = 2$