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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0:40 minutes

Problem 92

Textbook Question

Simplify each power of i. i^26

Verified step by step guidance

Understand that the imaginary unit \(i\) is defined as \(i = \sqrt{-1}\), and powers of \(i\) cycle every four: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\).

To simplify \(i^{26}\), find the remainder when 26 is divided by 4, since the powers of \(i\) repeat every 4.

Perform the division: 26 divided by 4 gives a quotient of 6 and a remainder of 2.

The remainder tells us which power of \(i\) corresponds to \(i^{26}\). Since the remainder is 2, \(i^{26} = i^2\).

Recall that \(i^2 = -1\), so \(i^{26}\) simplifies to the same value as \(i^2\).

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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.

Modulo Operation

The modulo operation is used to find the remainder of a division. In the context of simplifying powers of 'i', we can use modulo 4 to determine which power in the cycle corresponds to a given exponent. For example, to simplify i^26, we calculate 26 mod 4, which equals 2, indicating that i^26 simplifies to i^2.