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 $a\,b\,c\,d\in \mathrm{\mathbb{Z}}$ and $\gcd \left(a\,b\right)\=1\=\gcd \left(c\,d\right)$, then $\gcd \left(ab\,cd\right)=1$.

In addition, the following simplifications are performed:

Nested powers ${\left(a^b\right)}^c$ are combined as $a^{bc}$ if $a\in \mathrm{\mathbb{Q}}$ and one of the following conditions is satisfied:

$b\in \mathrm{\mathbb{Q}}$ and $\left(a<0\mathrm{or}-1<b<1\right)$

$c\in \mathrm{\mathbb{Z}}$

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 $b\&gt;1$ is made power-free, i.e., if there is a rational number $a\&gt;1$ 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 $\mathrm{-1}$ are pulled out of all rational powers.

$\mathrm{combine}\left(2^a{2}^b\&comma;'\mathrm{power}'\right)$

$2^{a+b}$

$\mathrm{combine}\left(2^a{2}^b\&comma;'\mathrm{icombine}'\right)$

$2^{a+b}$

$\mathrm{combine}\left(4^a{6}^b{12}^c{5}^d\&comma;'\mathrm{power}'\right)$

$4^a{6}^b{12}^c{5}^d$

$\mathrm{combine}\left(4^a{6}^b{12}^c{5}^d\&comma;'\mathrm{icombine}'\right)$

$2^{2a+b+2c}{3}^{b+c}{5}^d$

$\mathrm{combine}\left(4^a{6}^b{12}^c{5}^{d}F\left(6^a{30}^c{7}^e\right)\&comma;'\mathrm{icombine}'\right)$

$2^{2a+b+2c}{3}^{b+c}{5}^{d}F\left(5^c{6}^{a+c}{7}^e\right)$

$\mathrm{combine}\left(\frac{4^a}{6^a}{\left(\frac{15}{2}\right)}^b\&comma;'\mathrm{icombine}'\right)$

${\left(\frac{3}{2}\right)}^{-a+b}{5}^b$

$\mathrm{combine}\left(\frac{8^{\frac{1}{2}}}{4{\left(\mathrm{sqrt}\left(2\right)\right)}^a}\&comma;'\mathrm{icombine}'\right)$

$2^{-\frac{1}{2}-\frac{a}{2}}$

$\mathrm{combine}\left(\frac{\mathrm{sqrt}\left(2\right)2^{-\frac{a}{2}}{3}^a}{3}\&comma;'\mathrm{icombine}'\right)$

${\left(\frac{9}{2}\right)}^{-\frac{1}{2}+\frac{a}{2}}$

$\mathrm{combine}\left(\frac{{144}^a}{{25}^a}\&comma;'\mathrm{icombine}'\right)$

${\left(\frac{12}{5}\right)}^{2a}$

$\mathrm{combine}\left({\left(-2\right)}^{2a}{\left(-\frac{1}{3}\right)}^{-a}\&comma;'\mathrm{icombine}'\right)$

${\left(\mathrm{-1}\right)}^a{12}^a$