Simplify each power of i. i²⁶

Verified step by step guidance

1. Recall that the imaginary unit \(i\) is defined such that \(i^2 = -1\).
2. Recognize that powers of \(i\) repeat in a cycle of 4: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\), then the pattern repeats.
3. To simplify \(i^{26}\), find the remainder when 26 is divided by 4, since the powers cycle every 4 steps.
4. Calculate \(26 \div 4\) which gives a quotient of 6 and a remainder of 2, so \(i^{26} = i^{4 \times 6 + 2} = (i^4)^6 \times i^2\).
5. Since \(i^4 = 1\), simplify to \(1^6 \times i^2 = i^2\), and recall 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 repeat in a cycle of four: i¹ = i, i² = -1, i³ = -i, and i⁴ = 1. This pattern repeats for higher powers, so simplifying i raised to any integer power involves finding the remainder when the exponent is divided by 4.

Modular Arithmetic for Exponent Simplification

Modular arithmetic helps simplify powers by reducing the exponent modulo 4 in the case of i. For example, to simplify i^26, compute 26 mod 4 = 2, so i^26 = i² = -1. This technique streamlines calculations involving cyclic patterns.