Question

8-92.
Describe how you would evaluate i" where n could be any integer.

Answer 1

To evaluate i^n for any integer n, divide n by 4 and use the remainder to determine the result following the cycle: i, -1, -i, 1. This repeats every fourth power because i^4 equals 1.

Step-by-step explanation:

To evaluate i^n where n is any integer, you can use the properties of the imaginary unit i. Recall that i is defined such that i^2 = -1. From this definition, it follows that i^3 = i^2 × i = -i, and i^4 = i^2 × i^2 = 1. Every exponent of i can be expressed in terms of one of these four possibilities, because the powers of i repeat in a cycle: i, -1, -i, 1.

For any given integer n, you divide n by 4 and use the remainder to determine the power of i.

• If the remainder is 0, i^n = 1.
• If the remainder is 1, i^n = i.
• If the remainder is 2, i^n = -1.
• If the remainder is 3, i^n = -i.

This method applies to all integers, positive or negative, and is a fundamental concept in complex number arithmetic.

Answer 2

Step-by-step explanation:

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