Re: Simplify

• To: mathgroup at smc.vnet.net
• Subject: [mg32034] Re: Simplify
• From: "Alan Mason" <swt at austin.rr.com>
• Date: Wed, 19 Dec 2001 04:29:23 -0500 (EST)
• References: <9vmrje\$hh7\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```<Matthias.Bode at oppenheim.de> wrote in message
news:9vmrje\$hh7\$1 at smc.vnet.net...
> Dear Colleagues,
>
> why should I not expect MATHEMATICA to (Full)Simplify
>
> (a^b)^(1/b) or a^b^(1/b) to a?
>

Matthias,
the reason is because they are not equal for general complex-valued a, b, c.
The following notebook illustrates how to make a rule to enforce the desired
simplification (which is correct only for a, b, c real), and also gives a
case where equality fails.  z^w is defined as Exp[w Log[z]], and Log has a
branch cut; this complication is the reason for the failure.

In[1]:=
test = (a^b)^(1/b)
Out[1]=
\!\(\((a\^b)\)\^\(1\/b\)\)
In[2]:=
FullSimplify[%]
Out[2]=
\!\(\((a\^b)\)\^\(1\/b\)\)
In[3]:=
test2 = (a^b)^c
Out[3]=
\!\(\((a\^b)\)\^c\)
In[4]:=
FullSimplify[%]
Out[4]=
\!\(\((a\^b)\)\^c\)
In[5]:=
rule = Power[Power[a_, b_], c_]\[RuleDelayed] Power[a, b c]
Out[5]=
\!\(\((a_\^b_)\)\^c_ \[RuleDelayed] a\^\(b\ c\)\)
In[6]:=
test /. rule
Out[6]=
a
In[7]:=
test2 /. rule
Out[7]=
\!\(a\^\(b\ c\)\)
In[8]:=
a = 2 + 3 I;b = 3 +I; c = -2-4I;
In[9]:=
N[(a^b)^c]
Out[9]=
\!\(4.240023659792295`*^-7 - 7.75911100859049`*^-7\ \[ImaginaryI]\)
In[10]:=
N[a^(b c)]
Out[10]=
34864.2\[InvisibleSpace]-63800.3 \[ImaginaryI]

Alan

```

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