# With what exponent the power of any number becomes 0? Like we know that (any number)^0=1,so what will be the value of x in (any number)^x=0?

Jul 29, 2016

See below

#### Explanation:

Let $z$ be a complex number with structure

$z = \rho {e}^{i \phi}$ with $\rho > 0 , \rho \in \mathbb{R}$ and $\phi = a r g \left(z\right)$

we can ask this question. For what values of $n \in \mathbb{R}$ occurs

${z}^{n} = 0$ ?

Developping a little more

${z}^{n} = {\rho}^{n} {e}^{i n \phi} = 0 \to {e}^{i n \phi} = 0$

because by hypothese

$\rho > 0$.

So using Moivre's identity

${e}^{i n \phi} = \cos \left(n \phi\right) + i \sin \left(n \phi\right)$ then

${z}^{n} = 0 \to \cos \left(n \phi\right) + i \sin \left(n \phi\right) = 0 \to n \phi = \pi + 2 k \pi , k = 0 , \pm 1 , \pm 2 , \pm 3 , \cdots$

Finally, for

$n = \frac{\pi + 2 k \pi}{\phi} , k = 0 , \pm 1 , \pm 2 , \pm 3 , \cdots$

we get

${z}^{n} = 0$