Simplify each power of i. i¹¹

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 \(i^4 = 1\).
3. To simplify \(i^{11}\), find the remainder when 11 is divided by 4, since the powers repeat every 4.
4. Calculate \$11 \div 4\( which gives a quotient of 2 and a remainder of 3, so \)i^{11} = i^3$.
5. Use the cycle to identify \(i^3 = -i\), so \(i^{11}\) simplifies to \(-i\).

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 exponents modulo 4 in the case of i. For example, to simplify i¹¹, compute 11 mod 4 = 3, so i¹¹ = i³ = -i. This technique streamlines calculations involving cyclic patterns.