How do you evaluate i to large powers such as i^1992, i^1993, and i^2000 without a calculator?
since i^2 = -1, i^4 = 1.
So, divide the power by 4, since every factor of i^4 is just 1.
You will be left with one of the following:
i^0 = 1
i^1 = i
i^2 = -1
i^3 = -i
For example, i^913 = i^1 = i
To evaluate powers of complex numbers like i^1992, i^1993, and i^2000, we can use the properties of i. Let's break it down step by step:
1. Start by noting that i has a repeating pattern in its powers: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. After that, the pattern repeats.
2. For i to any power that is a multiple of 4, the result will always be 1. So, to evaluate i^2000, we can simply find i^4 because 2000 is divisible by 4. Therefore, i^2000 = 1.
3. To evaluate i^1992, notice that 1992 is divisible by 4, but there is a remainder of 2. This means we cannot directly use the pattern i^4 = 1. Instead, we can express i^1992 in terms of a multiple of 4 and the remainder: i^1992 = (i^4)^498 * i^2.
4. We already know from step 2 that i^4 = 1, so (i^4)^498 = 1^498 = 1.
5. The remaining factor, i^2, is easy to calculate. We know that i^2 = -1, so i^1992 = 1 * (-1) = -1.
6. Similarly, for i^1993, since there is a remainder of 1 when divided by 4, we can express it as i^1993 = (i^4)^498 * i^1.
7. As found earlier, (i^4)^498 = 1, and i^1 = i. Thus, i^1993 = 1 * i = i.
Therefore, the evaluations of the given expressions without using a calculator are as follows:
i^1992 = -1,
i^1993 = i, and
i^2000 = 1.