Question

Simplify: $i^{21}, i^{43},$ and $i^{52}$

Name: Problem 74, Chapter 2: Precalculus: Graphs and Models
Uploaded: 2020-06-21T07:04:14-08:00
Duration: 1 min 19 s
Channel: Jeyasree R T
Description: Simplify: i^21, i^43, and i^52

   Simplify: $i^{21}, i^{43},$ and $i^{52}$

Precalculus: Graphs and Models

Raymond A. Barnett,… 3rd Edition

Chapter 2, Problem 74

Instant Answer

Solved by Expert Jeyasree R T

06/21/2020

Step 1

These four values repeat in a cycle for every power of $i$. Show more…

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Simplify: $i^{21}, i^{43},$ and $i^{52}$

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Key Concepts

Periodicity of Powers of i

The powers of i follow a cyclic pattern with a period of 4, meaning that i^n repeats its values every 4 exponents. This cyclic behavior is crucial when simplifying high powers of i.

Modular Arithmetic in Exponentiation

Modular arithmetic allows one to reduce large exponents by taking them modulo 4 when dealing with powers of i. This technique simplifies the calculation by mapping any exponent to one of the four cases in the cyclic pattern.

Imaginary Unit

The imaginary unit, denoted by i, is defined by the property i² = -1. It serves as the fundamental building block for complex numbers, allowing operations with square roots of negative numbers.

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