Solve II=sqrt{1+xy}

$$I I = \sqrt { 1 + x y }$$

Solve for x

$$\left\{\begin{matrix}x\in \mathrm{R}\text{, }&y=0\text{ and }|I|=1\\x=\frac{I^{4}-1}{y}\text{, }&y\neq 0\end{matrix}\right.$$

Steps for Solving Radical Equations

Multiply I and I to get I^{2}.

$$I^{2}=\sqrt{1+xy}$$

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

$$\sqrt{1+xy}=I^{2}$$

Square both sides of the equation.

$$yx+1=I^{4}$$

Subtract 1 from both sides of the equation.

$$yx+1-1=I^{4}-1$$

Subtracting 1 from itself leaves 0.

$$yx=I^{4}-1$$

Divide both sides by y.

$$\frac{yx}{y}=\frac{I^{4}-1}{y}$$

Dividing by y undoes the multiplication by y.

$$x=\frac{I^{4}-1}{y}$$

Solve for I (complex solution)

$$I=-\sqrt[4]{xy+1}$$

$$I=\sqrt[4]{xy+1}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{I^{4}-1}{y}\text{, }&y\neq 0\\x\in \mathrm{C}\text{, }&\left(I=i\text{ or }I=-i\text{ or }I=1\text{ or }I=-1\right)\text{ and }y=0\end{matrix}\right.$$

Solve for I

$$I=\sqrt[4]{xy+1}$$

$$I=-\sqrt[4]{xy+1}\text{, }\left(x\leq -\frac{1}{y}\text{ or }y\geq 0\right)\text{ and }\left(x\geq -\frac{1}{y}\text{ or }y\leq 0\right)$$