Simplify each power of i. i^-13

Verified step by step guidance

1. Recall that the imaginary unit \(i\) is defined such that \(i^2 = -1\).
2. Understand that powers of \(i\) cycle every 4 steps: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then the pattern repeats.
3. To simplify \(i^{-13}\), rewrite the negative exponent as a reciprocal: \(i^{-13} = \frac{1}{i^{13}}\).
4. Find the remainder when 13 is divided by 4 to use the cyclic pattern: calculate \(13 \mod 4\).
5. Use the remainder to express \(i^{13}\) in terms of \(i^r\) where \(r\) is the remainder, then write \(i^{-13} = \frac{1}{i^{r}}\) and simplify further if possible.

Key Concepts

Imaginary Unit i and its Powers

The imaginary unit i is defined as the square root of -1, with the property i² = -1. Powers of i cycle every four steps: i¹ = i, i² = -1, i³ = -i, and i⁴ = 1, then the pattern repeats. Understanding this cycle is essential for simplifying powers of i.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, a⁻ⁿ = 1/aⁿ. Applying this to powers of i means i⁻¹ = 1/i, which can be further simplified using properties of i.

Modular Arithmetic for Exponent Reduction

Since powers of i repeat every 4, exponents can be reduced modulo 4 to simplify calculations. For example, i^n = i^(n mod 4). This technique helps simplify large positive or negative exponents by finding their remainder when divided by 4.