Textbook Question
Find each sum or difference. Write answers in standard form. (-3+2i) - (-4+2i)
1. Equations & Inequalities
The Imaginary Unit
3:58 minutesProblem 63a
Textbook Question
Write each complex number in standard form. (2 + 3i)3
Verified step by step guidance1
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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3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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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 form is crucial for clarity and consistency in mathematical communication, especially when performing further calculations or comparisons.
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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 De Moivre's theorem. In this case, (2 + 3i)^3 requires expanding the expression, which involves calculating the coefficients of the resulting terms. Mastery of exponentiation is vital for simplifying complex expressions and converting them into standard form.
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