Simplify each power of i. i^-27

The imaginary unit 'i' is defined as the square root of -1. It is a fundamental concept in complex numbers, allowing for the extension of real numbers to include solutions to equations that do not have real solutions, such as x^2 + 1 = 0. Powers of 'i' cycle through four values: i^0 = 1, i^1 = i, i^2 = -1, and i^3 = -i.

The powers of 'i' exhibit a cyclical pattern every four exponents. Specifically, i^n can be simplified by finding the remainder of n when divided by 4. This means that i^4 = 1, i^5 = i, i^6 = -1, and i^7 = -i, and this pattern continues indefinitely. Understanding this cycle is crucial for simplifying expressions involving powers of 'i'.

Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent. For example, i^-n = 1/(i^n). This concept is essential for simplifying expressions with negative powers, as it allows us to convert them into a more manageable form, often leading to the application of the cyclical nature of powers of 'i' for simplification.