Ch. 1 - Equations and Inequalities

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

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 53

Chapter 2, Problem 53

In Exercises 53–60, write each power of i as as i, - 1, - i, or 1. i³¹

Verified step by step guidance

Step 1: Recall the cyclical nature of powers of i. The powers of i repeat in a cycle of four: 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^31, divide the exponent (31) by 4 and find the remainder. This is because the cycle repeats every 4 powers.

Step 3: Perform the division: 31 ÷ 4. The quotient is 7, and the remainder is 3. This means i^31 is equivalent to i^3.

Step 4: Refer back to the cycle of powers of i. From the cycle, i^3 = -i.

Step 5: Conclude that i^31 simplifies to -i based on the cyclical pattern 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 and Division

To simplify powers of i, we can use the modulus of the exponent with respect to 4. For example, to find i^31, we calculate 31 mod 4, which gives us a remainder of 3. This means i^31 is equivalent to i^3, allowing us to simplify the expression effectively.

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 powers of i, as they form the basis for many operations in algebra involving imaginary units.

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Related Practice

Textbook Question

Solve each equation in Exercises 41–60 by making an appropriate substitution. $x^{\frac{2}{5}} + x^{\frac{1}{5}} - 6 = 0$

Textbook Question

In Exercises 48–57, perform the indicated operations and write the result in standard form. 6/(5+i)

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/(2x - 2) + 1/2 = 2/(x - 1)

Textbook Question

Solve each equation in Exercises 47–64 by completing the square. $x^{2} + 4 x + 1 = 0$

Textbook Question

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

In Exercises 53–60, write each power of i as as i, - 1, - i, or 1. i31

Verified video answer for a similar problem:

Key Concepts

Powers of i

Modulus and Division

Complex Numbers

In Exercises 53–60, write each power of i as as i, - 1, - i, or 1. i³¹