How do you simplify #7i^40 - 9i^100#?

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Answer 1 · 2015-11-10T11:22:23

#7i^40 - 9i^100 = -2#

We will use two facts.

1) #x^(ab) = (x^a)^b#

2) #i^2 = -1 => i^4 = (i^2)^2 = 1#

Applying (1), we get

#7i^40 - 9i^100 = 7i^(4*10) - 9i^(4*25) = 7(i^4)^10 - 9(i^4)^25#

Then, by (2),

#7(i^4)^10 - 9(i^4)^25 = 7*1^10 - 9*1^25 = 7 - 9 =-2#

So our final result is

#7i^40 - 9i^100 = -2#