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When I try to rationalize the following number $$1\over{2^{1/4}+4^{1/4}+8^{1/4}}$$

   FullSimplify[1/( 2^(1/4)+4^(1/4)+8^(1/4) )]

I get the same expression, and not my hand-calculation result which is

$${(\sqrt{4+3\sqrt{2}}-\sqrt 2) (3\sqrt 2 -2)}\over 14$$ What command should I use, if there is one?

Edit: "rationalize" meaning as in ordinary algebra where roots are moved from denominator to numerator, and not as writing a decimal as a fraction

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You can use ToRadicals and RootReduce instead:

Simplify @ ToRadicals @ RootReduce[1/(2^(1/4)+4^(1/4)+8^(1/4))] //TeXForm

$\frac{1}{14} \left(-6+2 \sqrt{2}+\sqrt{2 \left(8+9 \sqrt{2}\right)}\right)$

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    $\begingroup$ Very nice! Though we should make it clear to the OP that, while the solutions offered here work for his problem, none of them are general. Consider, for instance: test = {1/(Sqrt[2] + Sqrt[3] + Sqrt[5]), (3 + Sqrt[11])/(4 + Sqrt[11]), (2 + Sqrt[3])/(1 + Sqrt[5 + Sqrt[11]]), Sqrt[(1 + Sqrt[2])/(1 + Sqrt[3])], Sqrt[3 + 2 Sqrt[2]]/(1 + Sqrt[2])}; Given that this is not an obscure problem, yet MMA nevertheless doesn't have a built-in function for this, I gather it must be difficult to create a solution that is sufficiently general for Wolfram to offer it. $\endgroup$ – theorist Aug 4 '18 at 1:47
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    $\begingroup$ ....Though, having said that, there is such a function in Maple: maplesoft.com/support/help/maple/view.aspx?path=rationalize $\endgroup$ – theorist Aug 8 '18 at 0:27
  • $\begingroup$ @theorist Thanks for the link to Maple. It actually gives answer for variables a,b,c, but oddly enough answer does not look symmetric! $\endgroup$ – Maesumi Aug 9 '18 at 21:05
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In this case ToNumberField gives a denested form:

ToRadicals[ToNumberField[1/(2^(1/4) + 4^(1/4) + 8^(1/4))]] // Together // TeXForm

$\frac{1}{14} \left(-6+4 \sqrt[4]{2}+2 \sqrt{2}+2^{3/4}\right)$

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