Question

$$Q=AL^{\alpha}K^{1-\alpha}$$

Solve for A

$\left\{\begin{matrix}A=\frac{QK^{\alpha -1}}{L^{\alpha }}\text{, }&\left(L\neq 0\text{ and }K\neq 0\right)\text{ or }\left(L\neq 0\text{ and }\alpha =1\right)\text{ or }\left(\alpha =0\text{ and }K\neq 0\right)\\A\in \mathrm{C}\text{, }&\left(K=0\text{ or }L=0\right)\text{ and }\left(K=0\text{ or }\alpha \neq 0\right)\text{ and }\left(\alpha \neq 1\text{ or }L=0\right)\text{ and }Q=0\end{matrix}\right.$

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Solve for K

$\left\{\begin{matrix}K=e^{\frac{-Im(\alpha )arg(\frac{Q}{AL^{\alpha }})-iRe(\alpha )arg(\frac{Q}{AL^{\alpha }})+iarg(\frac{Q}{AL^{\alpha }})}{\left(Re(\alpha )\right)^{2}+\left(Im(\alpha )\right)^{2}-2Re(\alpha )+1}+\frac{2\pi n_{1}iRe(\alpha )}{\left(Re(\alpha )\right)^{2}+\left(Im(\alpha )\right)^{2}-2Re(\alpha )+1}+\frac{2\pi n_{1}Im(\alpha )}{\left(Re(\alpha )\right)^{2}+\left(Im(\alpha )\right)^{2}-2Re(\alpha )+1}-\frac{2\pi n_{1}i}{\left(Re(\alpha )\right)^{2}+\left(Im(\alpha )\right)^{2}-2Re(\alpha )+1}}\times \left(\frac{|Q||\frac{1}{L^{\alpha }}|}{|A|}\right)^{\frac{-Re(\alpha )+1+iIm(\alpha )}{\left(Re(\alpha )\right)^{2}+\left(Im(\alpha )\right)^{2}-2Re(\alpha )+1}}\text{, }n_{1}\in \mathrm{Z}\text{, }&AL^{\alpha }\neq 0\text{ or }\left(\alpha =0\text{ and }A\neq 0\right)\\K\in \mathrm{C}\text{, }&\left(\alpha \neq 0\text{ or }A=0\right)\text{ and }\left(L=0\text{ or }A=0\right)\text{ and }Q=0\end{matrix}\right.$