Solve (xy)^3=x^3y^3

$$( x y ) ^ { 3 } = x ^ { 3 } y ^ { 3 }$$

Solve for x (complex solution)

$$x\in \mathrm{C}$$

Solution Steps

Expand \left(xy\right)^{3}.

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

Subtract x^{3}y^{3} from both sides.

$$x^{3}y^{3}-x^{3}y^{3}=0$$

Combine x^{3}y^{3} and -x^{3}y^{3} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{C}$$

Solve for y (complex solution)

$$y\in \mathrm{C}$$

Solution Steps

Expand \left(xy\right)^{3}.

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

Subtract x^{3}y^{3} from both sides.

$$x^{3}y^{3}-x^{3}y^{3}=0$$

Combine x^{3}y^{3} and -x^{3}y^{3} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any y.

$$y\in \mathrm{C}$$

Solve for x

$$x\in \mathrm{R}$$

Solution Steps

Expand \left(xy\right)^{3}.

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

Subtract x^{3}y^{3} from both sides.

$$x^{3}y^{3}-x^{3}y^{3}=0$$

Combine x^{3}y^{3} and -x^{3}y^{3} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$

Solve for y

$$y\in \mathrm{R}$$

Solution Steps

Expand \left(xy\right)^{3}.

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

Subtract x^{3}y^{3} from both sides.

$$x^{3}y^{3}-x^{3}y^{3}=0$$

Combine x^{3}y^{3} and -x^{3}y^{3} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any y.

$$y\in \mathrm{R}$$