example of four exponentials conjecture

Taking $x_1=i\pi$, $x_2=i\pi \sqrt{2}$, $y_1=1$, $y_2=\sqrt{2}$, we see that this conjecture implies that one of $e^{i\pi }$, $e^{i\pi \sqrt{2}}$, or $e^{i2\pi }$ is transcendental. Since the first is $-1$ and the last is $1$, the conjecture states that second must be transcendental, that is, $e^{i\pi \sqrt{2}}$ is (conjecturally) transcendental.

In this particular case, the result is known already, so the conjecture is verified. Using Gelfond’s theorem, take $\alpha =e^{i\pi }$ and $\beta =\sqrt{2}$ and it follows that ${\alpha }^{\beta }$ is transcendental.