Problem 11 - Entrance Test

First, simplify the base of the expression, (1 + i) / (1 - i): To simplify, multiply the numerator and denominator by the conjugate of the denominator, which is (1 + i): (1 + i) / (1 - i) = ( (1 + i) * (1 + i) ) / ( (1 - i) * (1 + i) ) = (1 + 2i + i^2) / (1^2 - i^2) Since i^2 = -1: = (1 + 2i - 1) / (1 - (-1)) = (2i) / (1 + 1) = 2i / 2 = i. So the expression simplifies to i^2024. Now, we need to evaluate i^2024. The powers of i cycle with a period of 4: i^1 = i i^2 = -1 i^3 = -i i^4 = 1 i^5 = i, and so on. To find i^2024, we divide the exponent by 4 and look at the remainder. 2024 / 4 = 506 with a remainder of 0. When the remainder is 0, the power is equivalent to i^4, which is 1. So, i^2024 = (i^4)^506 = 1^506 = 1. The value of the expression is 1.