Domanda del libro di testo
In Exercises 48–57, perform the indicated operations and write the result in standard form. 6/(5+i)
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 2, Problema 53
In Exercises 53–60, write each power of i as as i, - 1, - i, or 1. i31
Guida verificata passo dopo passo1
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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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
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.
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05:10Higher Powers of i
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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04:22Dividing Complex Numbers
Pratica correlata
Domanda del libro di testo
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)
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Domanda del libro di testo
Solve each equation in Exercises 47–64 by completing the square.
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Domanda del libro di testo
Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? 1/R = 1/R1 + 1/R2 for R1
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Domanda del libro di testo
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. 2/(x - 2) = x/(x - 2) - 2
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Domanda del libro di testo
In Exercises 51–58, solve each compound inequality. - 3 ≤ x - 2 < 1
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