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I was looking at power functions. I am confused about the following:

The domain of the power function $x^{-1}$ is $\mathbb{R}\setminus\{0\}$. So take $x=-1$. Then why do I get the following contradiction?

$(-1)^{-1} = (-1)^{-\frac{2}{2}} = ((-1)^2)^{-\frac{1}{2}} = 1^{-\frac{1}{2}} = 1^{\frac{1}{2}} = 1$

I get the same contradiction for the discussion of any power function with an odd exponent. The same seems to be the case for the inverse function of a power function with odd exponent, i.e. the root function, since we do not have to restrict the domain, right? What am I missing. Which step is wrong and why? Please help. Thank you.

laguna
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  • There are 2 solutions for $1^\frac12$, which are $+1$ and $-1$. – PC1 Nov 28 '21 at 19:32
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    You'll presumably get a detailed answer, but the incorrect step is the second equality, which reads $-1 = 1$. The moral is not to expect $a^{bc} = (a^b)^c$ unless $0 < a$, or $b$ and $c$ satisfy conditions. – Andrew D. Hwang Nov 28 '21 at 19:32
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    In fact, the "rule" $a^{bc}=(a^b)^c$ is broken on much more elementary samples, such as $-1=(-1)^1=(-1)^{2\cdot\frac{1}{2}}=\sqrt{(-1)^2}=\sqrt{1}=1$ (??!) –  Nov 28 '21 at 19:41
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    You'll find many similar questions under the "fake-proofs" tag: https://math.stackexchange.com/questions/tagged/fake-proofs?tab=Frequent – Hans Lundmark Nov 28 '21 at 20:36
  • Thanks for the answers. The links helped me understand. – laguna Nov 28 '21 at 20:42

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