To evaluate i¹⁸, i¹⁹, i²⁰ and i²¹, we need to understand the cyclical nature of powers of 'i'. The powers of 'i' follow a repeating pattern: i, -1, -i, 1 and so on.
1. For i¹⁸, since the power is 18, we can divide it by 4 and find the remainder. The remainder of 18 divided by 4 is 2. That corresponds to i² = -1
2. For i¹⁹, the remainder of 19 divided by 4 is 3. That corresponds to i³ = -i
3. For i²⁰, the remainder of 20 divided by 4 is 0. That corresponds to i⁴ = 1
4. For i²¹, the remainder of 21 divided by 4 is 1. That corresponds to i¹ = i
Thus, our values for each complex number are i¹⁸ = -1, i¹⁹ = -i, i²⁰ = 1, and i²¹ = i.
When all these numbers are added, we get:
i¹⁸ + i¹⁹ + i²⁰ + i²¹ = -1 -i + 1 + i = 0
So, the answer is 0.