If you count $(-1)^1$ normally, it is just -1. But if you substitute 1 with 2/2, then it is $(-1)^{2/2}$ and if you separate 2/2 into $2\cdot 1/2$ then it is $((-1)^2)^{1/2}$. If you count this you will end up with $√1= 1$. Why does this happen? Can you not apply the laws of exponents this way?
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In the middle of that you used an exponent property $a^{bc}=(a^b)^c$. This is only valid if $a$ is nonnegative. And a few other special circumstances. So there is no justification for the side of your argument that ends with value $1$.
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+1 for an easily understood answer to a very frequently asked question on this site. – Ethan Bolker Oct 31 '24 at 15:00
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Hmm... perhaps we need an entry for $a^{bc} = (a^b)^c$ on the list of "Common Questions" post on meta. It's not quite like other exponentiation entries on that list. – Lee Mosher Oct 31 '24 at 15:25
\cdotinstead of • – jjagmath Oct 31 '24 at 15:01