How do I rationalize higher index roots? Often I've found exercises like

$$\frac{2-\sqrt[6]{3x+64}}{5x}$$

Where I need the rationalize. But I am not sure how to do it with indexes greater than $2$. How do I rationalize higher index roots? (the above one is just an example).

That is, perform

$$\frac{2-\sqrt[6]{3x+64}}{5x} \cdot \frac{2+\sqrt[6]{3x+64}}{2+\sqrt[6]{3x+64}}$$
(See my comments on original for links that explain this in detail... effectively, if we have a sum or difference involving an $n^{th}$ root we take either the sum of difference of $n^{th}$ roots and evaluate, using the given terms as the first factor and then plug in values for the remaining factor... it's messy, but it does work)
$$a^6 - b^6 = (a-b)(a^5 + a^4b + a^3b^2 + a^2b^3 + ab^4 + b^5 )$$
We now set $a = 2$ and $b =\sqrt[6]{3x+64}$ 
$$\frac{2-(3x+64)^{1/6}}{5x} * \frac{(3x+64)^{5/6}+2 (3 x+64)^{2/3}+4 \sqrt{3 x+64}+8 (3 x+64)^{1/3}+16 (3 x+64)^{1/6}+32}{(3x+64)^{5/6}+2 (3 x+64)^{2/3}+4 \sqrt{3 x+64}+8 (3 x+64)^{1/3}+16 (3 x+64)^{1/6}+32}$$

This yields
$$\frac{-3x}{5x[(3x+64)^{5/6}+2 (3 x+64)^{2/3}+4 \sqrt{3 x+64}+8 (3 x+64)^{1/3}+16 (3 x+64)^{1/6}+32]}$$

I hope you're happy... all that TeXing :P
the answer is
$-\frac{3x}{5x(8+\sqrt{3x+64})(4+2\sqrt[6]{3x+64}+\sqrt[3]{3x+64})}$