Show that $8 ^ \sqrt3\in\Bbb R$, and $(-8)^\sqrt3\not\in\Bbb R$.
closed as off-topic by John Doe, Arturo Magidin, anomaly, Saucy O'Path, Paul Frost Apr 27 at 23:39
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x^y = exp(y*ln(x))
(-8)^(sqrt3) = exp((sqrt3)*ln(-8))
ln(-8) is not in real, hence exp((sqrt3)*ln(-8)) is also not in real.