Re: [Mathematica] simplification question of (1/x)^n * x^n ----> 1 ?

"Nasser Abbasi" <nma@xxxxxxxxx> writes:


"Bhuvanesh" <BhuvaneshBhatt@xxxxxxxxx> wrote in message
news:6285672.1191198802693.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxxxxx
(1/x)^n*x^n is not always 1.

In[1]:= (1/x)^n*x^n /. {x->-1, n->Pi}

2 Pi
Out[1]= (-1)

In[2]:= N[%]

Out[2]= 0.629682 + 0.776853 I

Bhuvanesh,
Wolfram Research

Thanks. It seems the thing that makes the difference is if one cay carry
the
exponent inside the brackets or not.

When doing this by hand, when I see (a/b)^n, I can rewrite it as (a^n /
b^n) right? at least I hope so :) I think this is called the dividing of
powers rule of something like that. (high school stuff.)

Fine if a and b are positive or n is an integer, but not true in general.
(a/b)^n = exp(n ln(a/b)), and ln(a/b) = ln(a) - ln(b) + 2 pi i m for some
integer m, so (a/b)^n = a^n/b^n exp(2 pi i m n).

So the question now is this: is (1/x)^Pi = 1/x^Pi ? According to
Mathematica these are NOT the same.

And again, Mathematica is quite correct. They are not the same when x is a
negative real (when using the principal branch).
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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