In Exercises 53–60, write each power of i as as i, - 1, - i, or 1.i^135

The imaginary unit i is defined as the square root of -1. Its powers cycle through four distinct values: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. This cyclical pattern repeats every four powers, which is crucial for simplifying higher powers of i.

To simplify powers of i, we can use the modulo operation. Specifically, we find the exponent modulo 4, since the powers of i repeat every four terms. For example, to simplify i^135, we calculate 135 mod 4, which helps determine the equivalent lower power of i.

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a and b are real numbers. Understanding complex numbers is essential for working with powers of i, as they form the basis of operations involving imaginary units.