Write each power of i as i, - 1, - i, or 1. i44

Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

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Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 55a

Chapter 2, Problem 55a

Write each power of i as i, - 1, - i, or 1.
i⁴⁴

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.

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Related Practice

Textbook Question

Exercises 41–60 contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. 3/(x + 2) + 2/(x - 2) = 8/(x + 2)(x - 2)

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Textbook Question

Write each power of i as i, - 1, - i, or 1.

i¹¹⁴

Textbook Question

Solve each equation in Exercises 41–60 by making an appropriate substitution. (x - 5)² - 4(x - 5) - 21 = 0

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Textbook Question

Solve each equation in Exercises 47–64 by completing the square. $x^{2} + 3 x - 1 = 0$

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Textbook Question

In Exercises 48–57, perform the indicated operations and write the result in standard form. √ (-32) - √ (-18)

Textbook Question

In Exercises 51–58, solve each compound inequality. - 3 ≤ (2/3)x - 5 < - 1