I was looking at some algebra cheat sheet, where it said that
something I never seen before:
(a^n)^(1/n) was "a" if n is odd, and |a| if n is even.
Here is the link
http://tutorial.math.lamar.edu/
It is in the "Algebra Cheat Sheet (Reduced)" little down the page there,
a small pdf file. at the bottom line.
I thought to try it on CAS to see what it says. Maple and
Maxima simplify (a^n)^(1/n) to "a" with no assumptions.
Mathematica will do it only if "n" is assumed to be integers
and also if "a" positive.
Maxima:
----------------
r:a^n$
radcan(r^(1/n));
a
Maple
-----------
restart;
r:=a^n:
simplify(r^(1/n),power,symbolic);
a
Mathematica:
------------ attempt 1---------------
In[27]:= Clear[a, n];
r = a^n;
Simplify[r^(1/n)]
Out[29]= (a^n)^(1/n)
------------ attempt 2---------------
In[30]:= Clear[a, n];
r = a^n;
Assuming[Element[n, Integers] && Element[a, Reals], Simplify[r^(1/n)]]
Out[32]= (a^n)^(1/n)
------------ attempt 3---------------
In[36]:= Clear[a, n];
r = a^n;
Assuming[a > 0, Simplify[r^(1/n)]]
Out[38]= (a^n)^(1/n)
------------ attempt 4---------------
In[33]:= Clear[a, n];
r = a^n;
Assuming[Element[n, Integers] && a > 0, Simplify[r^(1/n)]]
Out[35]= a
-----------------------------
Question is: Math wise, which CAS is correct? Is the cheat sheet
correct in saying
(a^n)^(1/n) was "a" if n is odd, and |a| if n is even.
What does your CAS do for this one?
--Nasser