icombine - Maple Help

combine/icombine

combine integer powers in a product

 Calling Sequence combine(e, 'icombine')

Parameters

 e - expression

Description

 • The combine(e,'icombine') function tries to combine products of powers of integers in such a way that the bases of the powers in the resulting product have no common factors. It does not, however, necessarily rewrite a product of integer powers as a product of prime powers.
 • More generally, powers of rational numbers are rewritten in such a way that if ${\left(\frac{a}{b}\right)}^{e}$ and ${\left(\frac{c}{d}\right)}^{f}$ are two distinct rational powers in the resulting product, with $a,b,c,d\in \mathrm{ℤ}$ and $\mathrm{gcd}\left(a,b\right)=1=\mathrm{gcd}\left(c,d\right)$, then $\mathrm{gcd}\left(ab,cd\right)=1$.
 • In addition, the following simplifications are performed:
 – Nested powers ${\left({a}^{b}\right)}^{c}$ are combined as ${a}^{bc}$ if $a\in \mathrm{ℚ}$ and one of the following conditions is satisfied:
 • $b\in \mathrm{ℚ}$ and $\left(a<0\phantom{\rule[-0.0ex]{1.0ex}{0.0ex}}\mathrm{or}\phantom{\rule[-0.0ex]{1.0ex}{0.0ex}}-1
 • $c\in \mathrm{ℤ}$
 – Powers whose exponents are rational multiples of each other are combined into a single power, whose base may be a rational number greater than $1$.
 – Every rational base $b>1$ is made power-free, i.e., if there is a rational number $a>1$ and a positive integer $n$ satisfying ${a}^{n}=b$, then $n=1$ and $a=b$.
 – Unless there is only a single power of a rational number and that base is negative, powers of $-1$ are pulled out of all rational powers.

Examples

 > $\mathrm{combine}\left({2}^{a}{2}^{b},'\mathrm{power}'\right)$
 ${{2}}^{{a}{+}{b}}$ (1)
 > $\mathrm{combine}\left({2}^{a}{2}^{b},'\mathrm{icombine}'\right)$
 ${{2}}^{{a}{+}{b}}$ (2)
 > $\mathrm{combine}\left({4}^{a}{6}^{b}{12}^{c}{5}^{d},'\mathrm{power}'\right)$
 ${{4}}^{{a}}{}{{6}}^{{b}}{}{{12}}^{{c}}{}{{5}}^{{d}}$ (3)
 > $\mathrm{combine}\left({4}^{a}{6}^{b}{12}^{c}{5}^{d},'\mathrm{icombine}'\right)$
 ${{2}}^{{2}{}{a}{+}{b}{+}{2}{}{c}}{}{{3}}^{{b}{+}{c}}{}{{5}}^{{d}}$ (4)
 > $\mathrm{combine}\left({4}^{a}{6}^{b}{12}^{c}{5}^{d}F\left({6}^{a}{30}^{c}{7}^{e}\right),'\mathrm{icombine}'\right)$
 ${{2}}^{{2}{}{a}{+}{b}{+}{2}{}{c}}{}{{3}}^{{b}{+}{c}}{}{{5}}^{{d}}{}{F}{}\left({{5}}^{{c}}{}{{6}}^{{a}{+}{c}}{}{{7}}^{{e}}\right)$ (5)
 > $\mathrm{combine}\left(\frac{{4}^{a}{\left(\frac{15}{2}\right)}^{b}}{{6}^{a}},'\mathrm{icombine}'\right)$
 ${\left(\frac{{3}}{{2}}\right)}^{{-}{a}{+}{b}}{}{{5}}^{{b}}$ (6)
 > $\mathrm{combine}\left(\frac{{8}^{\frac{1}{2}}}{{\left(\sqrt{2}\right)}^{a}\cdot 4},'\mathrm{icombine}'\right)$
 ${{2}}^{{-}\frac{{1}}{{2}}{-}\frac{{a}}{{2}}}$ (7)
 > $\mathrm{combine}\left(\frac{\sqrt{2}{2}^{-\frac{a}{2}}{3}^{a}}{3},'\mathrm{icombine}'\right)$
 ${\left(\frac{{9}}{{2}}\right)}^{{-}\frac{{1}}{{2}}{+}\frac{{a}}{{2}}}$ (8)
 > $\mathrm{combine}\left(\frac{{144}^{a}}{{25}^{a}},'\mathrm{icombine}'\right)$
 ${\left(\frac{{12}}{{5}}\right)}^{{2}{}{a}}$ (9)
 > $\mathrm{combine}\left({\left(-2\right)}^{2a}{\left(-\frac{1}{3}\right)}^{-a},'\mathrm{icombine}'\right)$
 ${\left({-1}\right)}^{{a}}{}{{12}}^{{a}}$ (10)