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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

Maesumi
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    LeafCount[1/(2^(1/4) + 4^(1/4) + 8^(1/4))] vs. LeafCount[(Sqrt[4 + 3 Sqrt[2]] - Sqrt[2]) (3 Sqrt[2] - 2)/14] might give you a hint why this happens. Moreover, "rationalize" is a very misleasing word here. – Henrik Schumacher Aug 03 '18 at 21:57
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    I don't know of a command for this. But people have written functions to do what you wish. You can find their solutions here: https://mathematica.stackexchange.com/questions/5283/rationalize-the-denominator-by-default https://mathematica.stackexchange.com/questions/9868/how-can-i-rationalize-the-denominator-of-an-expression – theorist Aug 03 '18 at 22:53

2 Answers2

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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

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

Carl Woll
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    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. – theorist Aug 04 '18 at 01:47
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    ....Though, having said that, there is such a function in Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=rationalize – theorist Aug 08 '18 at 00:27
  • @theorist Thanks for the link to Maple. It actually gives answer for variables a,b,c, but oddly enough answer does not look symmetric! – Maesumi Aug 09 '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

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

Greg Hurst
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