Solve x^y^z=x^y^z

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

Solve for x (complex solution)

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

Solution Steps

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

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

Combine x^{y^{z}} and -x^{y^{z}} 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

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

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

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

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any y.

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

Solve for x

$$\left\{\begin{matrix}x>0\text{, }&\left(y<0\text{ and }Denominator(z)\text{bmod}2=1\right)\text{ or }y>0\\x=0\text{, }&\left(y^{z}>0\text{ and }y<0\text{ and }Denominator(z)\text{bmod}2=1\right)\text{ or }y>0\\x<0\text{, }&\left(Denominator(y^{z})\text{bmod}2=1\text{ and }y<0\text{ and }Denominator(z)\text{bmod}2=1\right)\text{ or }\left(Denominator(y^{z})\text{bmod}2=1\text{ and }y>0\right)\\x\neq 0\text{, }&y=0\text{ and }z>0\end{matrix}\right.$$

Solution Steps

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

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

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

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any x.

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

Solve for y

$$\left\{\begin{matrix}y<0\text{, }&x>0\text{ and }Denominator(z)\text{bmod}2=1\\y\in \mathrm{R}\text{, }&\left(x<0\text{ and }Denominator(z)\text{bmod}2=1\text{ and }Denominator(y^{z})\text{bmod}2=1\text{ and }y<0\right)\text{ or }\left(x<0\text{ and }Denominator(y^{z})\text{bmod}2=1\text{ and }y>0\right)\text{ or }\left(x=0\text{ and }Denominator(z)\text{bmod}2=1\text{ and }y^{z}>0\text{ and }y<0\right)\\y=0\text{, }&z>0\text{ and }x\neq 0\\y>0\text{, }&x\geq 0\end{matrix}\right.$$

Solution Steps

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

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

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

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any y.

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