Simplify each power of i. i¹¹

Verified step by step guidance

1. Recall that the imaginary unit \( i \) has the property \( i^2 = -1 \). Powers of \( i \) cycle every 4 steps: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \), and then the pattern repeats.
2. To simplify \( i^{11} \), find the remainder when 11 is divided by 4, since the powers of \( i \) repeat every 4.
3. Calculate \( 11 \div 4 \) which gives a quotient of 2 and a remainder of 3, so \( 11 = 4 \times 2 + 3 \).
4. Rewrite \( i^{11} \) as \( i^{4 \times 2 + 3} = (i^4)^2 \times i^3 \).
5. Since \( i^4 = 1 \), simplify to \( 1^2 \times i^3 = i^3 \), and then use the cycle to express \( i^3 = -i \).

Key Concepts

Imaginary Unit i

The imaginary unit i is defined as the square root of -1, satisfying i² = -1. It is the fundamental building block for complex numbers and powers of i cycle through a pattern based on this definition.

Powers of i and Their Cyclic Pattern

Powers of i repeat every four exponents: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, and then the cycle repeats. This pattern helps simplify higher powers of i by reducing the exponent modulo 4.

Modular Arithmetic for Exponent Reduction

Modular arithmetic involves finding the remainder when dividing the exponent by 4 to simplify powers of i. For example, i¹¹ can be simplified by calculating 11 mod 4 = 3, so i¹¹ = i³ = -i.