Question

How many triplets (x, y, z) of positive real numbers can be found such that $x^y=z,y^z=xand{z}^x=y$?

Answer 1

$x^y=z$,$y^z=x$ and $z^x=y$

Put $x=y^z$ in $(x^y=z)$

$\Rightarrow y^{zy}=z,$

Now put $y=z^x$

$z^{xyz}=z$ $\Rightarrow xyz=1$

Hence only one value of $(x,y,z)$ is possible.