Complex number difficulties?

i^4 = 1, so you can look at the exponents mod 4. Your expression is the same as

(i^3 + 1/i)^4.

Now, i^3 = -i and 1/i = -i

So you have (-i - i)^4

That's (-2i)^4= (-2)^4 i^4 = 16.

i^4 = 1 and so i^60 = (i^4)^15 = 1^15 = 1

i^63 = i^3 = -i

Similarly i^104 = 1, and we can write 1/i^101 = i^3/ i^104 = -i

So now you see that (i^63 + 1/i^101)^4 = (-2i)^4 = 8

Regards – Ian H

i^63 = -i and i^101 = i, so the expression becomes

[ (-i + 1) / i ]^4. Now go inside the brackets and

multiply by i/i:

[ (1 - i) / -1 ]^4 = (i - 1)^4

= i^4 - 4i^3 + 6i^2 - 4i + 1

= 1 + 4i - 6 - 4i + 1

= -4