Rationalizing denominator of $\frac{7}{2+\sqrt{3}}$. Cannot match textbook solution I am given this expression and asked to simplify by rationalizing the denominator:
$$\frac{7}{2+\sqrt{3}}$$
The solution is provided:
$14 - 7\sqrt{3}$
I was able to get to this in the numerator but am left with a 4 in the denominator. Here are my steps:
$$\frac{7}{2+\sqrt{3}} * \frac{2-\sqrt{3}}{2-\sqrt{3}}$$
=
$$\frac{14 - 7\sqrt{3}}{4}$$
Presumably I should not have a denominator here since the solution I'm given is just whats in the numerator. Presumably my numerator calculation is correct, where did I go wrong on the denominator?

note that:
$$(a+b)(c+d)=ac+ad+bc+bd$$
so in some cases it simplifies to:
$$(a+b)(a-b)=a^2-b^2$$
for you, $a=2$ and $b=\sqrt{3}$ so $a^2-b^2=4-3=1$
