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Negative Equals Positive?

I was thinking about exponents of -1, and I came up with a contradiction. May anyone explain where the fallacy is in the following: \[(-1) ^ {\frac{3}{2}} = ((-1) ^ {3}) ^ {\frac{1}{2}} = \sqrt{(-1) ^ {3}} = \sqrt{-1} = i\] \[(-1) ^ {\frac{3}{2}} = ((-1) ^ {\frac{1}{2}}) ^ {3} = (\sqrt{-1}) ^ {3} = i ^ {3} = -i\] Therefore, \[i = -i\] Obviously, the conclusion is wrong. But I can't figure out where the fallacy lies. All the steps look quite legit to me. Thanks in advance for the help!

Note by Nilabha Saha
8 months, 2 weeks ago

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\(\sqrt{-1}=\pm i\).

Brilliant Member - 8 months, 2 weeks ago

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Thank you very much for providing the answer. I guess that sometimes the important minutiae just skip our brains, leading to problems.

Nilabha Saha - 8 months, 2 weeks ago

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