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- Feb 14, 2012

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Problem:

Rationalize the denominator of $\displaystyle \frac{1}{a^\frac{1}{3}+b^{\frac{1}{3}}+c^{\frac{1}{3}}}.$

I know that if we are asked to rationalize, say, something like $\displaystyle \frac{1}{1+2^{\frac{1}{3}}}$, what we could do is the following:

$\displaystyle \frac{1}{1+2^{\frac{1}{3}}}=\frac{1}{1+2^{\frac{1}{3}}}.\frac{1-2^{\frac{1}{3}}+2^{\frac{2}{3}}}{1-2^{\frac{1}{3}}+2^{\frac{2}{3}}}$

$\displaystyle \frac{1}{1+2^{\frac{1}{3}}}=\frac{1-2^{\frac{1}{3}}+2^{\frac{2}{3}}}{1-2^{\frac{1}{3}}+2^{\frac{2}{3}}+2^{\frac{1}{3}}-2^{\frac{2}{3}}+2}$

$\displaystyle \frac{1}{1+2^{\frac{1}{3}}}=\frac{1-2^{\frac{1}{3}}+2^{\frac{1}{3}}}{1+2}$

$\displaystyle \frac{1}{1+2^{\frac{1}{3}}}=\frac{1-2^{\frac{1}{3}}+2^{\frac{1}{3}}}{3}$

But this one seems like not to be the case with the problem that I am asking on this thread...

Could anyone give me some hints to tackle it?

Thanks in advance.