Question

$$A_{h}^{3}=\frac{1}{2}A^{4}h$$

Solve for A (complex solution)

$\left\{\begin{matrix}A=\sqrt[4]{2}ih^{-\frac{1}{4}}A_{h}^{\frac{3}{4}}\text{; }A=\sqrt[4]{2}h^{-\frac{1}{4}}A_{h}^{\frac{3}{4}}\text{; }A=-\sqrt[4]{2}h^{-\frac{1}{4}}A_{h}^{\frac{3}{4}}\text{; }A=-\sqrt[4]{2}ih^{-\frac{1}{4}}A_{h}^{\frac{3}{4}}\text{, }&h\neq 0\\A\in \mathrm{C}\text{, }&A_{h}=0\text{ and }h=0\end{matrix}\right.$

Solve for A_h (complex solution)

$A_{h}=\frac{2^{\frac{2}{3}}e^{\frac{2\pi i}{3}}\sqrt[3]{h}A^{\frac{4}{3}}}{2}$
$A_{h}=\frac{2^{\frac{2}{3}}\sqrt[3]{h}A^{\frac{4}{3}}}{2}$
$A_{h}=\frac{2^{\frac{2}{3}}e^{\frac{4\pi i}{3}}\sqrt[3]{h}A^{\frac{4}{3}}}{2}$

Solve for A

$\left\{\begin{matrix}A=\sqrt[4]{\frac{2A_{h}^{3}}{h}}\text{; }A=-\sqrt[4]{\frac{2A_{h}^{3}}{h}}\text{, }&\left(A_{h}\geq 0\text{ and }h>0\right)\text{ or }\left(A_{h}\leq 0\text{ and }h<0\right)\\A\in \mathrm{R}\text{, }&A_{h}=0\text{ and }h=0\end{matrix}\right.$

Solve for A_h

$A_{h}=\frac{2^{\frac{2}{3}}\sqrt[3]{h}A^{\frac{4}{3}}}{2}$