Simplify the cubic root of cube assuming real and positive variable

Question

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

gs1=-((Sqrt[3] (1/(1 + \[Alpha]^3))^(1/3))/2^(
  2/3)) + (r^3 (1 + Cos[\[Theta]] Sin[\[Theta]])^(3/2) + (
   3 Sqrt[3]
     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

Answer 1

This is on version 12, windows 10.

Another workaround is to force collect on r^3 on the second term

term1 = -((Sqrt[3] (1/(1 + α^3))^(1/3))/2^(2/3));
term2 = (r^3 (1 + Cos[θ] Sin[θ])^(3/2) + (3 Sqrt[3] r^3 (-1 + α^3) (Cos[θ] + Sin[θ]) Sin[
        2 θ])/(4 (1 + α^3)));
gs1 = term1 + Collect[term2, r^3]^(1/3)

Assuming[r > 0, Simplify[gs1]]

A quick experiment shows that M does not collect factors out when one term has something in the denominator. Compare

ClearAll[a, b, r];
term = (r^3 b +  (r^3) /2)^(1/3);
Assuming[r > 0, Simplify@term]

The above did not work. But this works

ClearAll[a, b, r];
term = (r^3 b + r^3 )^(1/3);
Assuming[r > 0, Simplify@term]

So to force r^3 to be factored out, one can use Collect

ClearAll[a, b, r];
term = (r^3 b + r^3/2 );
term = Collect[term, r^3]^(1/3);
Assuming[r > 0, Simplify@term]

And this is basically what I did for your more complicated example.

Answer 2

Wrapping gs1 with ExpandAll:

Simplify[ExpandAll @ gs1, Assumptions -> r > 0]

(-Sqrt[3] (1/(1 + α^3))^(1/3) +
r ((6 Sqrt[3] (-1 + α^3) Cos[θ]^2 Sin[θ] +
6 Sqrt[3] (-1 + α^3) Cos[θ] Sin[θ]^2 + 4 (1 + α^3) (1 + Cos[θ] Sin[θ])^(3/2)) /
( 1 + α^3))^(1/3))/2^(2/3)

TeXForm[%]

$\frac{r \sqrt[3]{\frac{6 \sqrt{3} \left(\alpha ^3-1\right) \sin (\theta ) \cos ^2(\theta )+6 \sqrt{3} \left(\alpha ^3-1\right) \sin ^2(\theta ) \cos (\theta )+4 \left(\alpha ^3+1\right) (\sin (\theta ) \cos (\theta )+1)^{3/2}}{\alpha ^3+1}}-\sqrt{3} \sqrt[3]{\frac{1}{\alpha ^3+1}}}{2^{2/3}}$

You can also use ReplaceAll to force the transformations you want:

gs1  /. Power[a_. x_^k_ + b_. x_^k_, c_] :> x ^(k c) (a + b)^c

TeXForm @ %

$r \sqrt[3]{\frac{3 \sqrt{3} \left(\alpha ^3-1\right) \sin (2 \theta ) (\sin (\theta )+\cos (\theta ))}{4 \left(\alpha ^3+1\right)}+(\sin (\theta ) \cos (\theta )+1)^{3/2}}-\frac{\sqrt{3} \sqrt[3]{\frac{1}{\alpha ^3+1}}}{2^{2/3}}$