Simplify each power of i. i^-14

Verified step by step guidance

1. Recall that the powers of the imaginary unit \(i\) cycle every four: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\).
2. To simplify \(i^{-14}\), first find the equivalent positive exponent by using the property \(i^{-n} = \frac{1}{i^n}\).
3. Calculate \(14 \mod 4\) to find the remainder when 14 is divided by 4, which helps determine the equivalent power of \(i\) within its cycle.
4. Since \(14 \mod 4 = 2\), \(i^{14} = i^2\).
5. Thus, \(i^{-14} = \frac{1}{i^2}\). Use the value of \(i^2 = -1\) to simplify further.

Key Concepts

Imaginary Unit (i)

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.

Powers of 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. For example, i^4 = 1, i^5 = i, and so forth. This cyclical nature is crucial for simplifying higher powers of 'i' efficiently.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent. For instance, i^-n = 1/(i^n). This concept is essential for simplifying expressions involving negative powers, allowing us to rewrite i^-14 as 1/(i^14) and then apply the cyclical pattern of powers of 'i' to find the simplified result.