Solve z=sqrt{x^3y}

$$z = \sqrt { x ^ { 3 } y }$$

Solve for y

$$\left\{\begin{matrix}y=\frac{z^{2}}{x^{3}}\text{, }&z\geq 0\text{ and }x\neq 0\\y\in \mathrm{R}\text{, }&z=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Radical Equations

Swap sides so that all variable terms are on the left hand side.

$$\sqrt{x^{3}y}=z$$

Square both sides of the equation.

$$x^{3}y=z^{2}$$

Divide both sides by x^{3}.

$$\frac{x^{3}y}{x^{3}}=\frac{z^{2}}{x^{3}}$$

Dividing by x^{3} undoes the multiplication by x^{3}.

$$y=\frac{z^{2}}{x^{3}}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{z^{\frac{2}{3}}}{\sqrt[3]{y}}\text{, }&z\geq 0\text{ and }y\neq 0\\x\in \mathrm{R}\text{, }&y=0\text{ and }z=0\end{matrix}\right.$$