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

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

• My scientific calculator shows it's value. Your's calculator must be programmed for evaluating powers of positive numbers only.
– V.G
Sep 2, 2020 at 5:15
• Every nonzero complex number has three cube roots. Looks like your calculator's favourite cube root of $-10$ isn't the same as your favourite one. Sep 2, 2020 at 5:17
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 $$\ln(10i^2)$$
It might be giving one of the complex roots of the number for some reason. For example $$x^3 = -8$$ has three solutions $$-2, 1-i\sqrt{3}, 1+i\sqrt{3}$$.