Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 2: Problem 52
For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{(1+3 i)(2-4 i)}{(1+2 i)} $$
Short Answer
Expert verified
The expression simplifies to \(3.6 - 5.2i\).
Step by step solution
01
Expand the Numerator
First, we need to expand the expression in the numerator: \((1 + 3i)(2 - 4i)\). Use the distributive property (FOIL method):\[(1)(2) + (1)(-4i) + (3i)(2) + (3i)(-4i)\]This simplifies to:\[2 - 4i + 6i - 12i^2\]Since \(i^2 = -1\), replace \(-12i^2\) by \(12\):\[2 - 4i + 6i + 12\]Combine like terms: \(2 + 12 = 14\) and \(-4i + 6i = 2i\)Thus, the expanded numerator is:\(14 + 2i\)
02
Simplify the Expression with the Denominator
Now, take the result from Step 1, \(14 + 2i\), and divide it by \(1 + 2i\).To simplify a complex division, multiply the numerator and the denominator by the conjugate of the denominator.The conjugate of \(1 + 2i\) is \(1 - 2i\). Multiply:\[\frac{(14 + 2i)(1 - 2i)}{(1 + 2i)(1 - 2i)}\]
03
Multiply by the Conjugate
Expand both the numerator and the denominator separately.For the numerator \((14 + 2i)(1 - 2i)\), use FOIL:\[(14)(1) + (14)(-2i) + (2i)(1) + (2i)(-2i)\]This yields:\(14 - 28i + 2i - 4i^2\).Since \(i^2 = -1\), \(-4i^2\) becomes \(4\). Thus:\[14 - 28i + 2i + 4\]Combine like terms: \(14 + 4 = 18\) and \(-28i + 2i = -26i\)Resulting in: \(18 - 26i\).
04
Simplify the Denominator
Expand the denominator \((1 + 2i)(1 - 2i)\) using the difference of squares formula:\[(1)^2 - (2i)^2 = 1 - 4i^2\]Since \(i^2 = -1\), this becomes:\(1 + 4 = 5\).
05
Divide the Expressions
Now, divide the simplified numerator \((18 - 26i)\) by the simplified denominator \(5\):\[\frac{18 - 26i}{5} = \frac{18}{5} - \frac{26}{5}i\].This yields the simplified complex number: \[3.6 - 5.2i\].
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Complex Conjugate
In the world of complex numbers, the **complex conjugate** is a pivotal concept. For any complex number of the form \(a + bi\), its complex conjugate is \(a - bi\). Essentially, you just change the sign of the imaginary part. This operation is useful for rationalizing complex numbers, especially when you're dividing them. When you multiply a complex number by its conjugate, the imaginary parts cancel out, leaving a real number. For example, with \((1 + 2i)\) and its conjugate \((1 - 2i)\), when multiplied together you get
- \((1^2) - (2i)^2\)
- which simplifies to \(1 + 4 = 5\) (because \(i^2 = -1\))
Imaginary Unit
The imaginary unit, denoted by \(i\), is fundamental in complex number arithmetic. It's defined by the relationship \(i^2 = -1\). This definition allows \(i\) to represent the square root of \(-1\), a concept that doesn't exist among real numbers. When performing operations with complex numbers, you'll often find yourself using this property. For instance, in the expression expansion
- \((3i)(-4i) = -12i^2\)
- since \(i^2 = -1\)
- it simplifies to \(12\).
FOIL Method
The FOIL method is a technique used to expand expressions like \((a + bi)(c + di)\). It stands for First, Outer, Inner, Last, referring to a mnemonic for multiplying two binomials. In our problem, we expanded both the numerator and the denominator using this method. Let's break it down with \((1 + 3i)(2 - 4i)\):
- First: \(1 \times 2 = 2\)
- Outer: \(1 \times -4i = -4i\)
- Inner: \(3i \times 2 = 6i\)
- Last: \(3i \times -4i = -12i^2\)
Simplified Complex Number
The goal of evaluating complex number expressions is often to write the result as a simplified complex number. A **simplified complex number** is typically expressed in the form \(a + bi\) where \(a\) and \(b\) are real numbers. To achieve this in our exercise, we divide the simplified expanded forms of the numerator and the denominator. This involves several steps, prominently:
- expanding using FOIL
- applying the complex conjugate to rationalize the denominator
- simplifying the fraction \(\frac{18 - 26i}{5}\)
- which results in: \(3.6 - 5.2i\)
