Answers:
[tex]i^{157} = i\\\\i^{315} = -i\\\\i^{102} = -1\\\\i^{76} = 1\\\\[/tex]
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Explanation:
By definition, [tex]i = \sqrt{-1}[/tex]
Squaring both sides gets us [tex]i^2 = -1[/tex]
Then multiply both sides by i to get [tex]i^3 = -i[/tex]
Repeat the last step and you should get [tex]i^4 = -i^2 = -(-1) = 1[/tex]
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Notice we have this pattern going on:
[tex]i^0 = 1\\\\i^1 = i\\\\i^2 = -1\\\\i^3 = -i\\\\i^4 = 1\\\\[/tex]
Once we reach i^4, we start the process over again.
It repeats every 4 terms.
This means we'll divide the exponent over 4 and look at the remainder. We ignore the quotient completely.
157/4 = 39 remainder 1
That remainder 1 is the exponent of the simplified term
[tex]i^{157} = i^1 = i[/tex]
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Similarly,
315/4 = 78 remainder 3
So [tex]i^{315} = i^3 = -i[/tex]
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102/4 = 25 remainder 2
[tex]i^{102} = i^2 = -1[/tex]
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76/4 = 19 remainder 0
[tex]i^{76} = i^0 = 1[/tex]
