Textbook Question
Solve each equation in Exercises 47–64 by completing the square.
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 2, Problem 60
Write each power of i as i, - 1, - i, or 1. i135
Verified step by step guidance1
Understand that the powers of i (the imaginary unit) follow a repeating cycle: i, -1, -i, 1. This cycle repeats every 4 powers.
To determine the value of i^135, divide the exponent 135 by 4 and find the remainder. This is because the cycle repeats every 4 powers.
Perform the division: 135 ÷ 4. The quotient is 33, and the remainder is 3. The remainder determines the position in the cycle.
Match the remainder to the cycle: A remainder of 1 corresponds to i, 2 corresponds to -1, 3 corresponds to -i, and 0 corresponds to 1.
Since the remainder is 3, i^135 corresponds to the third position in the cycle, which is -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.
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Powers of i
Modulo Operation
To simplify powers of i, we can use the modulo operation. Specifically, we find the exponent modulo 4, since the powers of i repeat every four terms. For example, to simplify i^135, we calculate 135 mod 4, which helps determine the equivalent lower power of i.
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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 of operations involving imaginary units.
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Related Practice
Textbook Question
In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |x| = 8
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Textbook Question
Solve each equation in Exercises 47–64 by completing the square.
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Textbook Question
In Exercises 59–94, solve each absolute value inequality. |x| < 3
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Textbook Question
Solve each equation in Exercises 41–60 by making an appropriate substitution.
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Textbook Question
In Exercises 59–94, solve each absolute value inequality. |x - 1| ≤ 2
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