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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2:01 minutes

Problem 55

Textbook Question

In Exercises 53–60, write each power of i as as i, - 1, - i, or 1.i^44

Verified step by step guidance

Step 1: Recall the powers of i follow a repeating cycle of 4: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. This cycle repeats for higher powers of i.

Step 2: To determine the value of i^44, divide the exponent (44) by 4 to find the remainder. This will help identify where in the cycle the power falls.

Step 3: Perform the division: 44 ÷ 4 = 11 with a remainder of 0. The remainder determines the position in the cycle.

Step 4: Since the remainder is 0, i^44 corresponds to i^4 in the cycle. From the cycle, we know that i^4 = 1.

Step 5: Conclude that i^44 simplifies to 1 based on the repeating cycle of powers of i.

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Key Concepts

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

Powers of i

The imaginary unit i is defined as the square root of -1. Its powers cycle through four distinct values: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. This cyclical pattern repeats every four powers, which is crucial for simplifying higher powers of i.

Modulus Operation

To simplify powers of i, we can use the modulus operation with respect to 4, since the powers of i repeat every four terms. For example, to find i^44, we calculate 44 mod 4, which equals 0. This indicates that i^44 corresponds to i^0, which is equal to 1.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a and b are real numbers. Understanding complex numbers is essential for working with i, as it allows for the manipulation and interpretation of expressions involving imaginary units in various mathematical contexts.