I know this is very simple but I couldn't find a reasonable solution for it in the archive. It seems that my Mathematica does not take into account the assumptions when it wants to simplify the expression. Here is the thing
I have the following expression for
gs1=-((Sqrt (1/(1 + \[Alpha]^3))^(1/3))/2^( 2/3)) + (r^3 (1 + Cos[\[Theta]] Sin[\[Theta]])^(3/2) + ( 3 Sqrt r^3 (-1 + \[Alpha]^3) (Cos[\[Theta]] + Sin[\[Theta]]) Sin[ 2 \[Theta]])/(4 (1 + \[Alpha]^3)))^(1/3)
If you copy and paste the above expression in your Mathematica notebook you will see that it contains the cube root of
r^3, which upon assuming that r is positive must be simply
r. So I simplify it using the following command
gs2 = Simplify[gs1, Assumptions -> r > 0]
However, it doesn't do anything. This is very trivial and it should easily take
r^3 out of the cube root and make it
r but it doesn't.
I was thinking that maybe this is not the correct way of doing it so I tested it with a very simple expression
Simplify[Surd[x^3, 3], Assumptions -> x > 0]
and it gives
x as the result.
Following the answer of @Nasser, I found something strange.
ClearAll[a, b, r]; term = (r^3 b + (r^3)/2)^(1/3); Simplify[term, Assumptions -> r > 0]
this does not simplify the term, however, when I edit the term and eliminate the denominator by multiplying by
0.5, instead of dividing by
2, I get what I am looking for
ClearAll[a, b, r]; term = (r^3 b + (r^3)*0.5)^(1/3); Simplify[term, Assumptions -> r > 0]
Here is the screenshot also