Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
![### Complex Number Division in Rectangular Form
#### Problem Statement:
Evaluate the expression \(\frac{-3 - 2i}{4i}\) and write the result in the form \(a + bi\).
#### Solution Steps:
1. **Expression Simplification:**
Given expression: \(\frac{-3 - 2i}{4i}\)
2. **Multiply the Numerator and Denominator by the Complex Conjugate of the Denominator:**
- The Conjugate of \(4i\) is \(-4i\).
- Multiplying the numerator and denominator by \(-4i\):
\[
\frac{(-3 - 2i) \cdot (-4i)}{4i \cdot (-4i)}
\]
3. **Simplify the Denominator:**
\[
4i \cdot (-4i) = -16i^2 = -16(-1) = 16
\]
4. **Expand the Numerator:**
\[
(-3 - 2i)(-4i) = (-3)(-4i) + (-2i)(-4i) = 12i + 8i^2
\]
5. **Simplify the Numerator (cont.):**
- Substitute \(i^2 = -1\):
\[
12i + 8(-1) = 12i - 8
\]
6. **Combine Terms:**
\[
\frac{12i - 8}{16}
\]
7. **Separate Real and Imaginary Parts:**
\[
= \frac{-8}{16} + \frac{12i}{16} = -\frac{1}{2} + \frac{3i}{4}
\]
#### Final Result:
- **The real number \(a\) equals** \(\boxed{-\frac{1}{2}}\)
- **The real number \(b\) equals** \(\boxed{\frac{3}{4}}\)
#### Reference:
For a step-by-step tutorial on dividing complex numbers, watch the accompanying video. [Video Link]
**Submit Your Answer:**
- Enter the values of \(a\) and \(b\) in the corresponding boxes and click 'Submit Question'.
**Interactive Element:**
- After solving the problem](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F661e79ee-05f0-4794-8531-9b5261a2e126%2F38577148-d312-4d51-8e7e-233ec2d12a4c%2Fjz1im3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Complex Number Division in Rectangular Form
#### Problem Statement:
Evaluate the expression \(\frac{-3 - 2i}{4i}\) and write the result in the form \(a + bi\).
#### Solution Steps:
1. **Expression Simplification:**
Given expression: \(\frac{-3 - 2i}{4i}\)
2. **Multiply the Numerator and Denominator by the Complex Conjugate of the Denominator:**
- The Conjugate of \(4i\) is \(-4i\).
- Multiplying the numerator and denominator by \(-4i\):
\[
\frac{(-3 - 2i) \cdot (-4i)}{4i \cdot (-4i)}
\]
3. **Simplify the Denominator:**
\[
4i \cdot (-4i) = -16i^2 = -16(-1) = 16
\]
4. **Expand the Numerator:**
\[
(-3 - 2i)(-4i) = (-3)(-4i) + (-2i)(-4i) = 12i + 8i^2
\]
5. **Simplify the Numerator (cont.):**
- Substitute \(i^2 = -1\):
\[
12i + 8(-1) = 12i - 8
\]
6. **Combine Terms:**
\[
\frac{12i - 8}{16}
\]
7. **Separate Real and Imaginary Parts:**
\[
= \frac{-8}{16} + \frac{12i}{16} = -\frac{1}{2} + \frac{3i}{4}
\]
#### Final Result:
- **The real number \(a\) equals** \(\boxed{-\frac{1}{2}}\)
- **The real number \(b\) equals** \(\boxed{\frac{3}{4}}\)
#### Reference:
For a step-by-step tutorial on dividing complex numbers, watch the accompanying video. [Video Link]
**Submit Your Answer:**
- Enter the values of \(a\) and \(b\) in the corresponding boxes and click 'Submit Question'.
**Interactive Element:**
- After solving the problem
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