$(-10)^\frac{1}{3}$ comes out to be imaginary

Question

The expression $\displaystyle (-10)^\frac{1}{3}$ returns errror when calculated with scientific calculator returns an imaginary number. But it's power is $\dfrac{1}{\text{odd number}}$.

Answer 1

Most calculators utilize the $log$ function to calculate powers, using$$x^y=e^{y\ln x}$$ given that $e^x$ and $\ln x$ are calculated quickly thanks to the math (embedded) processor.

A negative $x$ is either treated directly (using $i^2=-1$) or it's made positive first, and the negative is dealt with after.

In your case, probably that $(-10)^{\frac13}$ is calculated as $e^{\frac13 \ln(-10)}$, and $\ln(-10)$ would be calculated as $\ln(10i^2)$

Answer 2

It might be giving one of the complex roots of the number for some reason. For example $x^3 = -8$ has three solutions $-2, 1-i\sqrt{3}, 1+i\sqrt{3}$.