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GALLERY: I 27 Simplify

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The above is a formal way of going about it, but for quick mental calculations, remembering that #i^4=1# allows you to "pull out" the greatest multiple of 4 possible from the exponent.
This leaves you with an exponent of 0, 1, 2, or 3 (any greater and there is another multiple of 4 to remove).
Then it is easy to work out that #i^0 = 1# , #i^1 = i# , #i^2 = -1# , and #i^3 = -i# .

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