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FullSimplify[((Sqrt[5] - 1)/(2*2^(1/4)))*Sqrt[Sqrt[2]*5^(1/4) - Sqrt[Sqrt[5] - 1]]]

returns

(-13 + 6*Sqrt[5] - 2*Sqrt[85 - 38*Sqrt[5]])^(1/4)

What are the steps to show this identity?

Edit: The work of FullSimplify here is just collecting all factors under the square root and multiplying out. But the difficult way is backwards: How to see that the result can be written as a product with a factor

(Sqrt[5] - 1)^2
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  • $\begingroup$ RootReduce may be helpful, I think. $\endgroup$
    – RungeC
    Jan 19, 2021 at 10:26
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    $\begingroup$ Honestly, whatever method Mathematica is using under the hood is not likely to be easily done by hand, so I question the desire to ask Mathematica for a step-by-step. $\endgroup$ Jan 19, 2021 at 16:39

1 Answer 1

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You can get a step by step solution using WolframAlpha. For this you type == at the beginning of a cell. Then you type your expression and hit Return. In the result, there is a button labeled "Step-by-step solution".

It looks like:

enter image description here

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    $\begingroup$ The Step-by-Step solution consists of rationalizing the denominator and is not really helpful $\endgroup$
    – Andreas
    Jan 19, 2021 at 16:28
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    $\begingroup$ I do not think you can get more info. FullSimplify is built with efficiency in mind and not with an educational purpose. $\endgroup$ Jan 19, 2021 at 17:20

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