Hi,

I need a bit of help solving a question on surds if anyone can push me in the right direction with it.

The question is:

I've started multiplying out the bottom row, and have got as far as this:

$\displaystyle \frac{4\sqrt{3}+3\sqrt{7}}{3\sqrt{3}+\sqrt{7}} * \frac{3\sqrt{3}-\sqrt{7}}{3\sqrt{3}-\sqrt{7}}$

I then proceed to multiplying out the bottom row;

$\displaystyle 3*3+3\sqrt{3}-3\sqrt{7}+3\sqrt{3}+\sqrt{3}*\sqrt{3}-\sqrt{21}+3\sqrt{7}+\sqrt{21}-7$

I then proceed to collecting the like terms;

$\displaystyle (9+3-7)+(-3\sqrt{7}+3\sqrt{7})+(-\sqrt{21}+\sqrt{21})+(3\sqrt{3}+3\sqrt{3})$

$\displaystyle = 5 + 6\sqrt{3}$

From here, I really can't see how to 'rationalise the denominator'.

I'm only working on the bottom row right now, but if anyone can tell me where I'm going wrong with this that would be great. I'm sure it's something simple, I'm just not seeing it.

Thanks in advance.