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Magazine Chapter 12: Problem 25
perform the indicated operations, expressing all answers in the form \(a+b j\) $$(1-j)^{3}$$
Short Answer
Expert verified
The result of \((1-j)^3\) is \(4-2j\).
Step by step solution
01
Expand the Expression Using Binomial Theorem
Recognize that \((1-j)^3\) can be expanded using the binomial theorem, where for any \((a+b)^n\), the terms are given by the sum: \[a^n + \binom{n}{1} a^{n-1} b + \binom{n}{2} a^{n-2} b^2 + \ldots + b^n\].Here, let \(a=1\) and \(b=-j\) with \(n=3\).
02
Calculate Each Term of the Binomial Expansion
Calculate each term of the binomial expansion:- The first term: \(1^3 = 1\)- The second term: \(-3j\) because \(\binom{3}{1}(1)^2(-j) = -3j\)- The third term: \(-3j^2\) because \(\binom{3}{2}(1)(-j)^2 = -3j^2\)- The fourth term: \(-j^3\) because \((-j)^3 = -j^3\).
03
Simplify the Terms Using Powers of \(j\)
Recall that \(j^2 = -1\) and \(j^3 = -j\). Using these, simplify:- \(-3j^2 = -3(-1) = 3\)- \(-j^3 = -(-j) = j\).
04
Combine and Simplify the Expression
Combine all the terms:1. From Step 2: \(1 - 3j + 3 + j\)2. Combine similar terms: - The real parts: \(1 + 3 = 4\) - The imaginary parts: \(-3j + j = -2j\)Thus, the expression becomes \(4 - 2j\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Theorem
The Binomial Theorem is a powerful tool for expanding an expression raised to a power. In its simplest form, it helps us to expand
- \((a + b)^n\)
- \((a + b)\)
- \(a = 1\),
- \(b = -j\),
- \(n = 3\),
Powers of Imaginary Unit
Understanding powers of the imaginary unit \(j\) is critical when working with complex numbers.
- The imaginary unit, \(j\), satisfies the equation \(j^2 = -1\). This property is foundational because it translates into other powers of \(j\).
- For instance, \(j^3\) becomes \(-j\), as shown by multiplying \(j^2\) by \(j\).
- \(j^0 = 1\)
- \(j^1 = j\)
- \(j^2 = -1\)
- \(j^3 = -j\)
- \(j^4 = 1\) again
Algebraic Operations
Algebraic operations with complex numbers require careful handling of both real and imaginary parts. When combining expressions:
- Add or subtract like terms separately for real and imaginary components.
- This means treating them similarly to polynomial operations, but keeping track of \(j\)'s powers is crucial.
- Collect all real number terms together and do the same for all imaginary terms to simplify.
- In the given solution, this results in combining real numbers as \(1 + 3 = 4\) and imaginary components as \(-3j + j = -2j\).
Expansion of Expressions
The expansion of expressions, especially those involving complex numbers, can significantly simplify algebraic operations.
- Instead of manually calculating \((1 - j)^3\) through direct multiplication, the binomial theorem allows us to expand it as a sum of multiple terms.
- Each of these individual terms can then be addressed separately by applying simple algebraic rules.
- By following the rules closely for each term's calculation鈥攍ike using binomial coefficients and powers鈥攜ou can clearly organize and simplify the overall expression.
- The result is more manageable, as seen when we combined terms to express the result as \(4 - 2j\).
