Simplify each power of i. i26

Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 92

Chapter 2, Problem 92

Simplify each power of i. i²⁶

Verified step by step guidance

Recall that the imaginary unit $i$ is defined such that $i^2 = -1$.

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.

To simplify $i^{26}$, find the remainder when 26 is divided by 4, since the powers cycle every 4 steps.

Calculate $26 \div 4$ which gives a quotient of 6 and a remainder of 2, so $i^{26} = i^{4 \times 6 + 2} = (i^4)^6 \times i^2$.

Since $i^4 = 1$, simplify to $1^6 \times i^2 = i^2$, and recall that $i^2 = -1$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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