Solve (A*B)^2=A^2*CD^3

$$( A \times B ) ^ { 2 } = A ^ { 2 } \times C D ^ { 3 }$$

Solve for A (complex solution)

$$\left\{\begin{matrix}\\A=0\text{, }&\text{unconditionally}\\A\in \mathrm{C}\text{, }&B=-\sqrt{C}D^{\frac{3}{2}}\text{ or }B=\sqrt{C}D^{\frac{3}{2}}\end{matrix}\right.$$

Steps by Finding Square Root

Expand \left(AB\right)^{2}.

$$A^{2}B^{2}=A^{2}CD^{3}$$

Subtract A^{2}CD^{3} from both sides.

$$A^{2}B^{2}-A^{2}CD^{3}=0$$

Reorder the terms.

$$A^{2}B^{2}-CA^{2}D^{3}=0$$

Combine all terms containing A.

$$\left(B^{2}-CD^{3}\right)A^{2}=0$$

Dividing by B^{2}-CD^{3} undoes the multiplication by B^{2}-CD^{3}.

$$A^{2}=\frac{0}{B^{2}-CD^{3}}$$

Divide 0 by B^{2}-CD^{3}.

$$A^{2}=0$$

Take the square root of both sides of the equation.

$$A=0 A=0$$

The equation is now solved. Solutions are the same.

$$A=0$$

Steps Using the Quadratic Formula

Expand \left(AB\right)^{2}.

$$A^{2}B^{2}=A^{2}CD^{3}$$

Subtract A^{2}CD^{3} from both sides.

$$A^{2}B^{2}-A^{2}CD^{3}=0$$

Reorder the terms.

$$A^{2}B^{2}-CA^{2}D^{3}=0$$

Combine all terms containing A.

$$\left(B^{2}-CD^{3}\right)A^{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute B^{2}-CD^{3} for a, 0 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$A=\frac{0±\sqrt{0^{2}}}{2\left(B^{2}-CD^{3}\right)}$$

Take the square root of 0^{2}.

$$A=\frac{0±0}{2\left(B^{2}-CD^{3}\right)}$$

Multiply 2 times B^{2}-CD^{3}.

$$A=\frac{0}{2B^{2}-2CD^{3}}$$

Divide 0 by 2B^{2}-2D^{3}C.

$$A=0$$

Solve for B (complex solution)

$$\left\{\begin{matrix}\\B=-\sqrt{C}D^{\frac{3}{2}}\text{; }B=\sqrt{C}D^{\frac{3}{2}}\text{, }&\text{unconditionally}\\B\in \mathrm{C}\text{, }&A=0\end{matrix}\right.$$

Solve for A

$$\left\{\begin{matrix}\\A=0\text{, }&\text{unconditionally}\\A\in \mathrm{R}\text{, }&\left(B=0\text{ and }C=0\text{ and }D=0\right)\text{ or }\left(C\geq 0\text{ and }D\geq 0\text{ and }|B|=\sqrt{CD^{3}}\right)\text{ or }\left(D\leq 0\text{ and }C\leq 0\text{ and }|B|=\sqrt{CD^{3}}\right)\end{matrix}\right.$$

Solve for B

$$\left\{\begin{matrix}B\in \mathrm{R}\text{, }&A=0\\B=-\sqrt{CD^{3}}\text{; }B=\sqrt{CD^{3}}\text{, }&A\neq 0\text{ and }D\leq 0\text{ and }C\leq 0\\B=-\sqrt{C}D^{\frac{3}{2}}\text{; }B=\sqrt{C}D^{\frac{3}{2}}\text{, }&A\neq 0\text{ and }C\geq 0\text{ and }D\geq 0\end{matrix}\right.$$