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!

#Algebra

Easy Math Editor

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Comments

$\sqrt{-1}=\pm i$ .

A Former Brilliant Member - 4 years, 3 months ago

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

Nilabha Saha - 4 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$