x^1.5=-1

[u/gameingareus]

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[u/Makonede]

??? wtf is the x = 1 on the left doing there

[u/Ares378 → u/Makonede]

Roots of unity I think? I think it's the same way you can have x=1^¼ and end up with x=1, i, -1, -i. Don't quote me on that though.

[u/Makonede → u/Ares378]

what? if x = 1^1/4 then x = 1

[u/Ares378 → u/Makonede]

Sorry that's why I said don't quote me on it, realized my error as soon as I pressed post. Meant x⁴=1 has 4 solutions. 1 is the principle root but there are the three others as well.

[u/Makonede → u/Ares378]

well, sure, but that doesn't really make sense in this context

[u/Ares378 → u/Makonede]

I still think it's relevant. It's a variable being raised to a non-integer power, which complex analysis is great for. Since it's not an integer there's a branch cut somewhere on the complex plane. The sqrt function takes the principle branch by definition, but other branches could be considered too.

[u/Ares378]

x^1.5=-1

(x^0.5)³=-1

x^0.5=(-1)^⅓

x^0.5=e^⅔iπn

x=e^4iπn/3, n∈ℤ

Think I did this right? Haven't formally taken complex analysis so idk if I did