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
$$\frac{2-\sqrt[6]{3x+64}}{5x} \cdot \frac{2+\sqrt[6]{3x+64}}{2+\sqrt[6]{3x+64}}$$
 A: (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
A: I might called the attempted operation rationalizing the numerator, rather than rationalizing---which suggests that the operation, like a rationalizing substitution, results in a rational expression.
Anyway, one can use the telescoping identity
$$(u - v)(u^{n - 1} + u^{n - 2} v + \cdots + u v^{n - 2} + v^{n - 1}) = u^n - v^n$$
to rationalize a numerator or denominator of the form $a \pm \sqrt[n]{b}$ (where $a$ and $b$ are themselves rational), by taking $$a = u, \quad v = \mp \sqrt[n] b,$$ which gives the rational expression $$a^n - (-1)^n b$$ on the right-hand side.
Note that (1) this technique generalizes the technique called "multiplying by the conjugate" mentioned in the question (which corresponds to the case $n = 2$, and (2) the resulting expressions are often messy---after all, one is multiplying by a radical expression with $n$ terms involving various powers. This technique is useful, however, for things like evaluating limits of the form $\frac{0}{0}$ of certain radical expressions.
A: the answer is
$-\frac{3x}{5x(8+\sqrt{3x+64})(4+2\sqrt[6]{3x+64}+\sqrt[3]{3x+64})}$

