I used Basic Math Assistant to calculate the product of two radicals:
In[24]:= Power[x, (3)^-1] Power[x^2, (3)^-1]
Out[24]= x^(1/3) (x^2)^(1/3)
It’s obvious that the result must be simplified to x. Unfortunately, Mma didn’t provide such simplification…
Then, I calculated the same product as the product of two powers:
In[25]:= x^(2/3)*x^(1/3)
Out[25]= x
The result is that it must be expected.
Why the pallet application produces such an issue?
Unfortunately this can not be simplified without getting into trouble for some numbers:
x^(1/3) ((x^2)^(1/3))
% /. x -> -1.0 - I
or, without complex numbers, just negative numbers:
x^(1/3) ((x^2)^(1/3))
% /. x -> -2.0
You can however use
PowerExpand[x^(1/3) ((x^2)^(1/3))]
to simplify such kind of expressions, but be aware it might not give the right result always:
PowerExpand converts (a b)^c to a^cb^c, whatever the form of c is.
PowerExpand also converts (a^b)^c to a^(b c), whatever the form of c is.
The transformations made by PowerExpand are correct in general only if c is an integer or a and b are positive real numbers.
How about :
Simplify[Power[x, (3)^-1] Power[x^2, (3)^-1], x > 0]
(* x *)
In[5]:= Reduce[Power[x, (3)^-1] Power[x^2, (3)^-1] == x, x, Reals]
Out[5]= x >= 0
Evidently, it’s not a problem to obtain the correct result. But, Mma submit a strange result, especially for some users, e.g. school students.
It’s interesting to see the full form of the expression:
In my opinion, the result have to be more rigorous, e.g. to highlight correct result for Reals and Complex numbers, separately.
Branch cuts are not a problem, they are a feature of mathematics.
Indeed, not a problem, a feature! Squaring and taking a cube root simply do not commute. Commutator is not equal to 0.
Use CubeRoot or Surd to get the real-valued roots
Also read:
tutorial/FunctionsThatDoNotHaveUniqueValues
that you can find in the documentation.
