Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Complex Numbers

College Algebra

1. Equations & Inequalities

Complex Numbers

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1:08 minutes

Problem 90

Textbook Question

Simplify each power of i. i^29

Verified step by step guidance

Recognize that the powers of the imaginary unit \(i\) cycle every four: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\).

To simplify \(i^{29}\), find the remainder when 29 is divided by 4, since the powers of \(i\) repeat every 4.

Calculate \(29 \div 4\) to find the quotient and remainder. The remainder will determine the equivalent power of \(i\).

The remainder of \(29 \div 4\) is 1, so \(i^{29} = i^1\).

Thus, \(i^{29}\) simplifies to \(i\).

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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, denoted as 'i', is defined as the square root of -1. It is a fundamental concept in complex numbers, allowing for the extension of the real number system to include solutions to equations that do not have real solutions, such as x^2 + 1 = 0.

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after reaching a certain value, known as the modulus. In the context of simplifying powers of 'i', we use modulus 4 to determine the equivalent lower power of 'i' by calculating the exponent modulo 4.