<s>[INST] Simplifying fractional surds I have this fractional surd:
$$\frac{5\sqrt{7}+4\sqrt{2}}{3\sqrt{7}+5\sqrt{2}}$$
I can calculate this with a calculator fairly easily obviously but what is the best tactic without one?
Thank you!
 [/INST] So first remember some key rules:


*

*$\sqrt{a} \times \sqrt{a}=a$

*$\sqrt{a} \times \sqrt{b}=\sqrt{ab}$ (which you can simplify usually)


(There are more but we only really need these!)
So a good technique to use in this question is called rationalising the denominator. Which basically means take the denominator of the surd and change the middle sign so $3\sqrt{7}+5\sqrt{2}$ turns into $3\sqrt{7}-5\sqrt{2}$. So we have:
$$\frac{5\sqrt{7}+4\sqrt{2}}{3\sqrt{7}+5\sqrt{2}} \times \frac{3\sqrt{7}-5\sqrt{2}}{3\sqrt{7}-5\sqrt{2}}$$
Which we can do as the $2^{nd}$ fraction is equal to $1$.
Looking at the top now we have $({5\sqrt{7}+4\sqrt{2}}) \times ({3\sqrt{7}-5\sqrt{2}})= 65-13\sqrt{14}$
Then doing the same thing with the bottom we get it just equal to $13$,
So putting this together we get: $$\frac{65-13\sqrt{14}}{13}= 5-\sqrt{14}$$
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