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Suppose I have a simple algebraic equation like:

ChebyshevT[4, p] == 0
1 - 8 p^2 + 8 p^4 == 0

and I want to solve for the term p^4 by simple rearrangement:

p^4 == -(1 - 8 p^2)/8

How do I do that in Mathematica ? And how can I then assign the solution to replace.

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I am not sure what you are aiming for but you could get what I think is your desired form by the following:

sol=Solve[1 - 8 p^2 + 8 p^4 == 0 /. p^4 -> u, u]
u/.sol[[1]]

This yields:

1/8 (-1 + 8 p^2)
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There are several ways to do that. For example, this:

    Clear[eq1, eq2, p];

eq1 = ChebyshevT[4, p] == 0;
eq2 = eq1 /. a_*p^4 -> a*x;
p^4 == Solve[eq2, x][[1, 1, 2]]
p^4 == 1/8 (-1 + 8 p^2)  

or this:

Clear[eq1, eq2, p];

eq1 = ChebyshevT[4, p] == 0;
eq2 = Map[Divide[#, -8] &, Map[Subtract[#, eq1[[1, 3]]] &, eq1]]
1/8 (-1 + 8 p^2) == p^4  
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