Simplify each power of i. i^-14

Verified step by step guidance

1. Recall that the imaginary unit \(i\) satisfies the property \(i^4 = 1\), which means powers of \(i\) repeat every 4 steps.
2. Rewrite the negative exponent using the property of exponents: \(i^{-14} = \frac{1}{i^{14}}\).
3. Simplify the denominator by reducing the exponent modulo 4: find the remainder when 14 is divided by 4, i.e., calculate \(14 \mod 4\).
4. Express \(i^{14}\) as \(i^{(4 \times 3) + 2} = (i^4)^3 \times i^2\), and since \(i^4 = 1\), this simplifies to \(1^3 \times i^2 = i^2\).
5. Substitute back to get \(i^{-14} = \frac{1}{i^2}\), and then simplify \(\frac{1}{i^2}\) using the fact that \(i^2 = -1\).

Key Concepts

Imaginary Unit i

The imaginary unit i is defined as the square root of -1, satisfying i² = -1. It is the fundamental unit used to extend the real number system to complex numbers, allowing for the representation and manipulation of numbers involving the square roots of negative values.

Powers of i and Their Cyclic Pattern

Powers of i follow a repeating cycle every four exponents: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, and then the pattern repeats. This cyclicity helps simplify any integer power of i by reducing the exponent modulo 4.

Negative Exponents

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