Simplify each power of i. i²⁹

Verified step by step guidance

1. Recall that the powers of the imaginary unit \(i\) cycle every 4 powers: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\).
2. To simplify \(i^{29}\), find the remainder when 29 is divided by 4, since the powers repeat every 4.
3. Calculate \(29 \div 4\) which gives a quotient and a remainder. The remainder determines the equivalent power of \(i\).
4. Express \(i^{29}\) as \(i^{4k + r}\) where \(k\) is the quotient and \(r\) is the remainder from the division.
5. Use the fact that \(i^{4k} = (i^4)^k = 1^k = 1\), so \(i^{29} = i^r\), and then substitute the value of \(r\) to find the simplified form.

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 of complex numbers and is used to extend the real number system to include solutions to equations like x² + 1 = 0.

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 this context. For example, to simplify i²⁹, compute 29 mod 4 = 1, so i²⁹ = i¹ = i. This technique streamlines calculations involving cyclic patterns.