Question

How do you rationalize and divide `(2+sqrt3) /(2-sqrt3)`?

Answer 1

Using the identity `(a+b)(a-b)=a^2 - b^2`, let's multiply and divide the fraction by `2+\sqrt{3}`. You'll notice that, when doing so, at the numerator you'll have `(2+\sqrt{3})^2`, while at the denominator you'll have `(2-\sqrt{3})(2+\sqrt{3})`, which is the formula mentioned at the beginning.

So, `(2-\sqrt{3})(2+\sqrt{3})=2^2 - (\sqrt(3))^2 = 4-3=1`.

Since the denominator is `1`, we can ignore it, and the final answer is thus `(2+\sqrt{3})^2`