Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 12: Problem 27
perform the indicated operations, expressing all answers in the form \(a+b j\) $$\frac{6 j}{2-5 j}$$
Short Answer
Expert verified
The complex number is \(-\frac{30}{29} + \frac{12}{29}j\).
Step by step solution
01
Identify the Complex Fraction
The given expression is a complex number division problem: \( \frac{6j}{2 - 5j} \). The goal is to express this in the form \(a + bj\).
02
Multiply by the Conjugate
To simplify \( \frac{6j}{2 - 5j} \), multiply both the numerator and the denominator by the conjugate of the denominator, \(2 + 5j\), to eliminate the imaginary part in the denominator. This gives:\[\frac{6j(2 + 5j)}{(2 - 5j)(2 + 5j)}\]
03
Calculate the Denominator Product
Calculate the product of the denominator using the formula \((a - b)(a + b) = a^2 - b^2\). Thus,\[(2 - 5j)(2 + 5j) = 2^2 - (5j)^2 = 4 - 25(-1) = 4 + 25 = 29\]
04
Calculate the Numerator Product
Expand the numerator:\[6j(2 + 5j) = 6j \times 2 + 6j \times 5j = 12j + 30j^2\]Since \(j^2 = -1\), we have:\[12j + 30(-1) = 12j - 30\]
05
Combine into the Standard Form
Combine the calculations to form the complex number:\[\frac{6j}{2 - 5j} = \frac{-30 + 12j}{29} = \frac{-30}{29} + \frac{12}{29}j\]Therefore, the expression in the form \(a + bj\) is \(-\frac{30}{29} + \frac{12}{29}j\).
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 Fraction
When dealing with complex numbers, a complex fraction occurs when we have a fraction where either the numerator, denominator, or both are complex numbers. In the original exercise, we have the complex fraction \( \frac{6j}{2 - 5j} \). These types of fractions can be tricky because they involve imaginary units, making direct simplification challenging.
To manage complex fractions:
To manage complex fractions:
- Convert the denominator to a real number by eliminating the imaginary component.
- Multiply by the conjugate of the denominator.
Conjugate Multiplication
In complex number operations, the conjugate plays a crucial role, especially in division and simplification. The conjugate of a complex number \( a + bj \) is \( a - bj \). Multiplying by the conjugate helps to eliminate the imaginary part of the denominator in a complex fraction. For example, the conjugate of \(2 - 5j\) is \(2 + 5j\).
Using conjugate multiplication:
Using conjugate multiplication:
- The product of a complex number and its conjugate is always real because it can be represented by the formula \((a + bj)(a - bj) = a^2 + b^2\).
- This property is leveraged to remove the imaginary component in the denominator of a fraction.
Imaginary Unit
The imaginary unit, denoted as \( j \), is a fundamental concept in complex numbers. It is defined by the property \( j^2 = -1 \). Understanding this property is vital when simplifying expressions involving complex numbers.
The role of the imaginary unit:
The role of the imaginary unit:
- In calculations, like in our exercise, you often encounter expressions like \( 30j^2 \), which simplifies to \( 30(-1) \) or \(-30\).
- It allows complex numbers to extend real numbers' capabilities, enabling solutions to equations that wouldn't have real solutions, like \( x^2 + 1 = 0 \).
Standard Form of a Complex Number
The standard form of a complex number is expressed as \( a + bj \), where \( a \) and \( b \) are real numbers, and \( j \) represents the imaginary unit. Presenting complex numbers in this form simplifies calculations and makes them more intuitive.
Use the standard form because:
Use the standard form because:
- It clearly separates the real and imaginary components of a number.
- Expressing answers in standard form, like \( -\frac{30}{29} + \frac{12}{29}j \), ensures consistency and clarity in communication.
