Domanda del libro di testo
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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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 63a
Write each complex number in standard form. (2 + 3i)3
Guida verificata passo dopo passo1
Rewrite the expression \((2 + 3i)^3\) using the binomial theorem: \((a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\), where \(a = 2\), \(b = 3i\), and \(n = 3\).
Expand the binomial using the formula: \((2 + 3i)^3 = \binom{3}{0}(2)^3(3i)^0 + \binom{3}{1}(2)^2(3i)^1 + \binom{3}{2}(2)^1(3i)^2 + \binom{3}{3}(2)^0(3i)^3\).
Simplify each term in the expansion: \(\binom{3}{0}(2)^3(3i)^0 = 8\), \(\binom{3}{1}(2)^2(3i)^1 = 36i\), \(\binom{3}{2}(2)^1(3i)^2 = -54\), and \(\binom{3}{3}(2)^0(3i)^3 = -27i\).
Combine the real and imaginary parts: Add the real terms \(8 - 54\) and the imaginary terms \(36i - 27i\).
Write the result in standard form \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient 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 is the real part, b is the imaginary part, and i is the imaginary unit defined as the square root of -1. Understanding complex numbers is essential for performing operations such as addition, subtraction, multiplication, and division, as well as for converting them into standard form.
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Dividing Complex Numbers
Standard Form of Complex Numbers
The standard form of a complex number is a + bi, where a and b are real numbers. To express a complex number in standard form, one must ensure that the imaginary unit i is isolated in the second term. This is particularly important when performing operations like exponentiation, as the result must be simplified to fit this format.
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05:02Multiplying Complex Numbers
Exponentiation of Complex Numbers
Exponentiation of complex numbers involves raising a complex number to a power, which can be done using the binomial theorem or by expanding the expression. In the case of (2 + 3i)^3, one must apply the distributive property or binomial expansion to calculate the result accurately. Understanding how to manipulate and simplify these expressions is crucial for arriving at the correct standard form.
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04:22Dividing Complex Numbers
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