Textbook Question
Simplify and write the result in standard form. √-108
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1. Equations & Inequalities
The Imaginary Unit
3:40 minutesProblem 62a
Textbook Question
Write each complex number in standard form. (1 - i)3
Verified step by step guidance1
Rewrite the expression \((1 - i)^3\) using the binomial theorem, which states \((a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\). Here, \(a = 1\), \(b = -i\), and \(n = 3\).
Expand the binomial expression: \((1 - i)^3 = \binom{3}{0}(1)^3(-i)^0 + \binom{3}{1}(1)^2(-i)^1 + \binom{3}{2}(1)^1(-i)^2 + \binom{3}{3}(1)^0(-i)^3\).
Simplify each term: \(\binom{3}{0}(1)^3(-i)^0 = 1\), \(\binom{3}{1}(1)^2(-i)^1 = -3i\), \(\binom{3}{2}(1)^1(-i)^2 = 3(-1) = -3\), and \(\binom{3}{3}(1)^0(-i)^3 = -i\).
Combine the simplified terms: \(1 - 3i - 3 - i\).
Group the real and imaginary parts to write the complex number in standard form \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of \(i\).
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Video duration:
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 manipulating and performing operations on them, such as addition, subtraction, multiplication, and exponentiation.
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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 is crucial when simplifying expressions involving complex numbers, as it allows for easier interpretation and further calculations.
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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 the case of (1 - i)^3, one can expand the expression using the binomial theorem, which states that (a + b)^n can be expressed as a sum of terms involving combinations of a and b raised to various powers. This process is key to simplifying complex expressions.
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