Step-by-step explanation:
remember, i = sqrt(-1)
you know what an exponent means, right ?
it means you multiply the enclosed term that many times with itself.
when you envision this as a long chain of (1+i)/(1-i) multiplications, you can see you can group them into pairs, and then multiply these pairs - just half the times than for the single terms.
so,
((1+i)/(1-i))²⁰²⁰ = ((1+i)/(1-i))² ^ 1010
to get an idea what the result of the long chain of multiplications might be, let's start with the square and see, if we can find a pattern, that we then can use to come to a conclusion about the total result.
((1+i)/(1-i))² = (1+i)²/(1-i)²
(1+i)² = 1×1 + 1×i + 1×i + i² = 1 + 2i - 1 = 2i
(1-i)² = 1×1 - 1×i - 1×i + i² = 1 - 2i - 1 = -2i
so,
((1+i)/(1-i))² = 2i/-2i = -1
aha !
so,
((1+i)/(1-i))² ^ 1010 = (-1)¹⁰¹⁰
just to be sure you understand, we use the same trick again
(-1)¹⁰¹⁰ = (-1)² ^ 505 = 1⁵⁰⁵ = 1
as 1 to any power is always 1.
so, the right answer is A.
remember, a negative number to an even power results always in a positive number. and to an uneven/odd power in a negative number.
again, because such an exponent is only the summary of many multiplications.
