If i^2=-1, then (1+i)^20-(1-i)^20 equals

1974 AMC 12 Problem 17

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17.

i^2=-1,

then

(1+i)^{20}-(1-i)^{20}

equals

-1024

-1024i

0

1024

1024i

Compute the fourth power of each conjugate base

Both

(1+i)^4

and

(1-i)^4

equal

-4

Since

(1+i)^2=2i,

we have

(1+i)^4=-4.

Likewise

(1-i)^2=-2i

and

(1-i)^4=-4.

Hence each twentieth power equals

(-4)^5=-1024,

and their difference is

0.

Therefore, the correct answer is C.