How do you simplify i^(-43)+i^(-32)i−43+i−32 ?

1 Answer
George C.
Apr 11, 2017

i^(-43)+i^(-32) = i+1i−43+i−32=i+1

Explanation:

Look at the first few non-negative powers of ii:

i^0 = 1i0=1

i^1 = ii1=i

i^2 = -1i2=−1

i^3 = -ii3=−i

i^4 = 1i4=1

Basically this pattern: 1, i, -1, -i1,i,−1,−i repeats every 44 powers.

In terms of angles, multiplying by ii is an anticlockwise rotation of pi/2π2 in the complex plane. So after 44 rotations we are back facing the same way.

So in general we can write:

{ (i^(4n) = 1), (i^(4n+1) = i), (i^(4n+2) = -1), (i^(4n+3) = -i) :}

which holds for any integer n.

Now:

-43 = -44+1 = 4(-11)+1

-32 = -32+0 = 4(-8)+0

So:

i^(-43)+i^(-32) = i+1

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