18 questions linked to/from What are the Laws of Rational Exponents?
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### Splitting the exponent of a negative number? [duplicate]

Recently I have been chewing on a bit of a paradox in my mind, and I'm trying to figure out what I'm doing wrong. I am not in high school anymore, I have just graduated from college, and I haven't ...
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### How does advancing through the math major work? [closed]

I am an undergrad math major that just completed Calculus 3 last semester. This semester I signed up for Discrete Mathematics, and will be taking Intro to Advanced/Abstract Math next. Of course-- I ...
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I am confused about something: \begin{eqnarray} (e^{2 i \pi})^{0.5} = (e^{2 i \pi \cdot 0.5})= e^{i \pi}=-1 \end{eqnarray} but \begin{eqnarray} e^{2 i \pi}=1~ and~ 1^{0.5}=1 \end{eqnarray} ...
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### Are complex numbers subject to different rules of math? [duplicate]

From what I know, the rule to distribute exponents is like: $$(a b)^x = a^x b^x$$ Thus, if $a = \sqrt 2$ and $b = \sqrt 3$, $ab = \sqrt 6$. However, the imaginary unit $i = \sqrt{-1}$ has a ...
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### Can the argument of a logarithm and its base be negative at the same time?

I'm struggling to understand why a $\log_x y$ is only true for $y > 0$. I know that if $y$ was $0$ , $x$ could only be $0$, but if it was $0$, then $y$ could be anything. So we'll leave that ...
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### Exponent Laws: Fractional Exponents and Negative Bases

I have read through this thread. It mentions that there a several conventions for rational exponents.The first condition for the first convention for exponent laws to apply is that the base be greater ...
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### Square root branch cut

Consider the following expression: $$\phi(\delta)=i\,\sqrt{-3+i\,\delta},$$ where $\delta$ is infinitesimal. If we choose a branch cut along the negative real axes, it ...
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### Why is $x^{p/q}$ ill-defined for $x<0$.

This is probably a duplicate but I can't find, if you do let me know and I will delete. Why is $x^{p/q}$ ill-defined for $x<0$. I can see that it is, $(-1)^{1/3} \neq (-1)^{2/6}$, but why? I ...
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### How to plot the equation$x(x^2-1)^\frac{1}{3}$?

I would like to plot the function $x(x^2-1)^\frac{1}{3}$. When I do it in a statistical computing package, I dont get anything between $-1$ and $1$. However, when I plot this on google, I get a ...
The question is, how can we know if we can solve a n-index root without imaginary numbers when n-index is a non integer number, but a real number? First, basic theory: $$\sqrt[2.0]{8} = 2.8284.....$$...
### $e^{i\theta}$ confusion [duplicate]
I learned the following $$e^{ik2\pi}=1$$ and I was wondering whether or not $k$ has to be an integer. Thought 1: Since $e^{ik2\pi}=\cos(2k\pi)+i\sin(2k\pi)=1$, equating the real and imaginary ...