Solve a^m/b^m=1

$$\frac { a ^ { m } } { b ^ { m } } = 1$$

Solve for a (complex solution)

$$a=e^{\frac{Im(m)arg(b^{m})+iRe(m)arg(b^{m})}{\left(Re(m)\right)^{2}+\left(Im(m)\right)^{2}}-\frac{2\pi n_{1}iRe(m)}{\left(Re(m)\right)^{2}+\left(Im(m)\right)^{2}}-\frac{2\pi n_{1}Im(m)}{\left(Re(m)\right)^{2}+\left(Im(m)\right)^{2}}}\left(|b^{m}|\right)^{\frac{Re(m)-iIm(m)}{\left(Re(m)\right)^{2}+\left(Im(m)\right)^{2}}}$$

$$n_{1}\in \mathrm{Z}$$

$$b\neq 0\text{ or }m=0$$

Solve for b (complex solution)

$$b=e^{\frac{Im(m)arg(a^{m})+iRe(m)arg(a^{m})}{\left(Re(m)\right)^{2}+\left(Im(m)\right)^{2}}-\frac{2\pi n_{1}iRe(m)}{\left(Re(m)\right)^{2}+\left(Im(m)\right)^{2}}-\frac{2\pi n_{1}Im(m)}{\left(Re(m)\right)^{2}+\left(Im(m)\right)^{2}}}\left(|a^{m}|\right)^{\frac{Re(m)-iIm(m)}{\left(Re(m)\right)^{2}+\left(Im(m)\right)^{2}}}$$

$$n_{1}\in \mathrm{Z}$$

$$m=0\text{ or }a\neq 0$$

Solve for a

$$\left\{\begin{matrix}a=\left(b^{m}\right)^{\frac{1}{m}}\text{, }&\left(Denominator(m)\text{bmod}2=1\text{ and }Numerator(m)\text{bmod}2=1\text{ and }b^{m}<0\text{ and }b\neq 0\text{ and }\left(b^{m}\right)^{\frac{1}{m}}\neq 0\right)\text{ or }\left(m\neq 0\text{ and }b<0\text{ and }Denominator(m)\text{bmod}2=1\text{ and }b^{m}>0\right)\text{ or }\left(\left(b^{m}\right)^{\frac{1}{m}}<0\text{ and }m\neq 0\text{ and }Denominator(m)\text{bmod}2=1\text{ and }b>0\right)\text{ or }\left(\left(b^{m}\right)^{\frac{1}{m}}<0\text{ and }b^{m}\geq 0\text{ and }m>0\text{ and }Denominator(m)\text{bmod}2=1\text{ and }b>0\right)\text{ or }\left(\left(b^{m}\right)^{\frac{1}{m}}>0\text{ and }b>0\text{ and }b^{m}\geq 0\text{ and }m>0\right)\text{ or }\left(\left(b^{m}\right)^{\frac{1}{m}}>0\text{ and }m\neq 0\text{ and }b>0\right)\\a=-\left(b^{m}\right)^{\frac{1}{m}}\text{, }&\left(b^{m}>0\text{ and }m\neq 0\text{ and }b<0\text{ and }Denominator(m)\text{bmod}2=1\text{ and }Numerator(m)\text{bmod}2=0\right)\text{ or }\left(m\neq 0\text{ and }b>0\text{ and }\left(b^{m}\right)^{\frac{1}{m}}>0\text{ and }Numerator(m)\text{bmod}2=0\text{ and }Denominator(m)\text{bmod}2=1\right)\text{ or }\left(b^{m}<0\text{ and }Denominator(m)\text{bmod}2=1\text{ and }Numerator(m)\text{bmod}2=1\text{ and }\left(b^{m}\right)^{\frac{1}{m}}>0\text{ and }b\neq 0\text{ and }Numerator(m)\text{bmod}2=0\right)\text{ or }\left(b^{m}<0\text{ and }b<0\text{ and }Denominator(m)\text{bmod}2=1\text{ and }Numerator(m)\text{bmod}2=1\text{ and }\left(b^{m}\right)^{\frac{1}{m}}<0\text{ and }Numerator(m)\text{bmod}2=0\right)\text{ or }\left(m\neq 0\text{ and }b>0\text{ and }\left(b^{m}\right)^{\frac{1}{m}}<0\text{ and }Numerator(m)\text{bmod}2=0\right)\\a\neq 0\text{, }&b\neq 0\text{ and }b\neq -1\text{ and }m=0\end{matrix}\right.$$

Solve for b

$$\left\{\begin{matrix}b=\left(a^{m}\right)^{\frac{1}{m}}\text{, }&\left(m\neq 0\text{ and }a^{m}>0\text{ and }Denominator(m)\text{bmod}2=1\text{ and }a<0\right)\text{ or }\left(a^{m}<0\text{ and }Denominator(m)\text{bmod}2=1\text{ and }Numerator(m)\text{bmod}2=1\text{ and }a<0\right)\text{ or }\left(\left(a^{m}\right)^{\frac{1}{m}}<0\text{ and }a>0\text{ and }a^{m}\geq 0\text{ and }m>0\text{ and }Denominator(m)\text{bmod}2=1\right)\text{ or }\left(\left(a^{m}\right)^{\frac{1}{m}}<0\text{ and }m\neq 0\text{ and }a>0\text{ and }Denominator(m)\text{bmod}2=1\right)\text{ or }\left(\left(a^{m}\right)^{\frac{1}{m}}>0\text{ and }a>0\text{ and }a^{m}\geq 0\text{ and }m>0\right)\text{ or }\left(\left(a^{m}\right)^{\frac{1}{m}}>0\text{ and }m\neq 0\text{ and }a>0\right)\text{ or }\left(Denominator(m)\text{bmod}2=1\text{ and }Numerator(m)\text{bmod}2=1\text{ and }a^{m}<0\text{ and }a>0\text{ and }\left(a^{m}\right)^{\frac{1}{m}}\neq 0\right)\\b=-\left(a^{m}\right)^{\frac{1}{m}}\text{, }&\left(a^{m}>0\text{ and }m\neq 0\text{ and }a<0\text{ and }Denominator(m)\text{bmod}2=1\text{ and }Numerator(m)\text{bmod}2=0\right)\text{ or }\left(a<0\text{ and }Denominator(m)\text{bmod}2=1\text{ and }Numerator(m)\text{bmod}2=1\text{ and }Numerator(m)\text{bmod}2=0\text{ and }a^{m}<0\right)\text{ or }\left(a^{m}<0\text{ and }a>0\text{ and }Numerator(m)\text{bmod}2=1\text{ and }\left(a^{m}\right)^{\frac{1}{m}}>0\text{ and }Denominator(m)\text{bmod}2=1\text{ and }Numerator(m)\text{bmod}2=0\right)\text{ or }\left(\left(a^{m}\right)^{\frac{1}{m}}>0\text{ and }m\neq 0\text{ and }a>0\text{ and }Numerator(m)\text{bmod}2=0\text{ and }Denominator(m)\text{bmod}2=1\right)\text{ or }\left(a^{m}<0\text{ and }a>0\text{ and }\left(a^{m}\right)^{\frac{1}{m}}<0\text{ and }Numerator(m)\text{bmod}2=0\text{ and }Numerator(m)\text{bmod}2=1\right)\text{ or }\left(m\neq 0\text{ and }a>0\text{ and }\left(a^{m}\right)^{\frac{1}{m}}<0\text{ and }Numerator(m)\text{bmod}2=0\right)\\b\neq 0\text{, }&m=0\text{ and }a\neq 0\end{matrix}\right.$$