How do you simplify i^21 + i^30i21+i30?

1 Answer
KillerBunny
Nov 13, 2015

i-1i−1

Explanation:

Observe that the powers of ii are cyclic:

i^0= 1i0=1
i^1 = ii1=i
i^2 = -1i2=−1
i^3=-ii3=−i
i^4 = 1i4=1
...

So, to find high powers of i, we can calculate the modulus 4 Since 21 = 5*4 +1, and 30 = 7*4 +2, we have that i^21=i^1=i, and i^30 = i^2 = -1. So,

i^21+i^30 = i-1.

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