Simplify each power of i. i^26

Key Concepts

Imaginary Unit (i)

The imaginary unit, denoted as 'i', is defined as the square root of -1. It is a fundamental concept in complex numbers, allowing for the extension of the real number system to include solutions to equations that do not have real solutions, such as x^2 + 1 = 0.

Powers of i

The powers of 'i' follow a cyclical pattern: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. This cycle repeats every four powers, which means any power of 'i' can be simplified by finding its equivalent within this cycle, making calculations more manageable.

Modulo Operation

The modulo operation is used to find the remainder of a division. In the context of simplifying powers of 'i', we can use modulo 4 to determine which power in the cycle corresponds to a given exponent. For example, to simplify i^26, we calculate 26 mod 4, which equals 2, indicating that i^26 simplifies to i^2.