Rationalize the Denominator?

Question

I don't know if my brain is fried from studying or if this is a really tricky problem, but I can't get it.
3 + sqrt(3) / 4-2sqrt(3)

According to my book, the answer is 9+5sqrt(3) / 2

Answer 1

3+sqrt3/4-2sqrt3 mult by 4+2sqrt3

(3+sqrt3)(4+2sqrt3) / 16-12

12 + 10sqrt3 + 6 / 4

18 + 10 sqrt3/ 4

9 + 5 sq rt 3 / 2

Answer 2

Multiply the numerator and denominator by the conjugate of the denominator: 4 + 2√3. You get this:

(12 + 10√3 + 6) / [16 - 4(3)] = (18 + 10√3) / (16 -12) =
(18 + 10√3) / 4 = 2 (9 + 5√3) / 2 (2) = (9 + 5√3) / 2

Answer 3

To rationalize a denominator means to take the given denominator, change the sign in front of it and multiply it by the numerator and denominator originally given.

This is done because we cannot have a square root in the denominator of a fraction.

We have this guy:

3 + sqrt(3) / 4-2sqrt(3)

Multiply the numerator and denominator by 4 + 2sqrt{3}.

Noticed that I changed the sign after 4 to POSITIVE?

Can you take it from here?

Guido

Answer 4

Multiple the numerator and denominator by 4+2sqrt(3). Expand and simplify each to get the answer.