# Why is $-1=e^{\pi i }=e^{2\pi i\frac{1}{2}}\neq (e^{2\pi i})^{\frac{1}{2}}=1^{\frac{1}{2}}=1$ true?

I thought there was this rule that $$e^{xB}=(e^x)^B$$?

Also what I don't understand is that $$x^{\frac{1}{2}}$$ is defined as the square root of $$x$$. And because the square root may also be in $$\mathbb{C}$$ it doesn't matter which number $$x$$ is because every complex number has a n-th root.

But for $$1^{\frac{1}{2}}$$ we have the roots $$-1$$ and $$1$$, so I could also write $$-1$$ on the right side then the Statement above would be true and not true at the same time.

But we said the Expression in the Question is correct and to write $$-1$$ is false. Can somebody explain the reason we choose one Version over another altough they are equivalent?

• To be precise every complex number $($beside $0$$)$ has exactly $n$ distinct $n$th roots. – mrtaurho Jan 27 at 12:07
• @mrtraurho Well, $0$ doesn't have distinct $n$th roots. – J.G. Jan 27 at 12:08
• Conclusion: using the notation $z^{1/2}$ to mean a single number and simultaneously to mean a set of two numbers, can lead to chaos. Should we be surprised? – Did Jan 27 at 12:10
• The main problem of taking a root in general is that it happens to be that the root function is a multivalued one. Thus, even within the reals you can state that $5=\sqrt{25}=-5$ which is a contradiction as well. – mrtaurho Jan 27 at 12:13
• Which rule? If your question is how to define a square root function on the complex plane, the answer is well known (and it has been hashed and rehashed on this site, I must add). – Did Jan 27 at 12:18

If one is working in real analysis, the expression $$x^{1/2}$$ ($$x>0$$) is by definition $$\sqrt{x}$$, which is defined to be the positive real number $$y$$ such that $$y^2=x$$. Hence one has $$1^{1/2}=1.$$
If one is working in complex analysis, $$1^{1/2}$$ can be viewed as the multivalued function $$f(z)=z^{1/2}$$ evaluated at $$z=1$$. In this context, since $$f$$ is multivalued, it is incorrect to write $$1^{1/2}=1$$.
The identity $$(e^x)^y = e^{xy}$$ holds for real numbers $$x$$ and $$y$$, but assuming its truth for complex numbers leads to paradox like the one you have observed.