/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 67 Perform the indicated operation ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 67

Perform the indicated operation and simplify. Write each answer in the form \(a+b i\) $$ \frac{2}{3-2 i} $$

Short Answer

Expert verified
\(\frac{6}{13} + \frac{4}{13}i\)

Step by step solution

01

- Identify the Conjugate

To rationalize the denominator, identify the conjugate of the denominator. The conjugate of \(3 - 2i\) is \(3 + 2i\).
02

- Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate obtained in Step 1: \[\frac{2}{3-2i} \times \frac{3+2i}{3+2i} = \frac{2(3+2i)}{(3-2i)(3+2i)}\]
03

- Distribute in the Numerator

Distribute the numerator: \[2(3 + 2i) = 6 + 4i\]
04

- Simplify the Denominator

Simplify the denominator using the difference of squares formula: \[(3-2i)(3+2i) = 9 - (2i)^2 = 9 - 4(-1) = 9 + 4 = 13\]
05

- Combine Numerator and Denominator

Combine the simplified numerator and denominator: \[\frac{6 + 4i}{13}\]
06

- Simplify the Fraction

Simplify the expression by dividing each part of the numerator by the denominator: \[\frac{6}{13} + \frac{4i}{13} = \frac{6}{13} + \frac{4}{13}i\]

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.

Rationalizing Denominators
Rationalizing the denominator involves eliminating any complex number in the denominator from a fraction. We do this because it makes computations easier and fractions more presentable. To rationalize a denominator in a fraction with a complex number, we need to multiply both the numerator and the denominator by the conjugate of the complex denominator.
Let's consider our example: \(\frac{2}{3-2i}\). Here, the denominator is the complex number \((3-2i)\).
The conjugate of \((3-2i)\) is \((3+2i)\). Next, we'll multiply the numerator and the denominator by this conjugate. Always remember, multiplying by the conjugate helps to convert a complex denominator into a real number.
Conjugate in Complex Numbers
A conjugate in complex numbers is formed by changing the sign between the real part and the imaginary part of the number. If you have a complex number \((a + bi)\), its conjugate is \((a - bi)\).
In our exercise, we initially had \((3 - 2i)\). Its conjugate is \((3 + 2i)\).
This method helps in transforming complex division into a simpler multiplication and addition problem using the difference of squares. Here's the important part: when a complex number is multiplied by its conjugate, the result is a real number.
  • For example, \((3 - 2i)(3 + 2i) = 9 + 4 = 13\).

Notice how the imaginary parts are effectively eliminated, leaving a nice real number.
Simplifying Complex Fractions
After multiplying and distributing, we end up with a new fraction, like \(\frac{6 + 4i}{13}\) in our case.
The goal is to express the answer in the form \((a + bi)\), where both parts are rational numbers. To achieve this, split the fraction: \(\frac{6 + 4i}{13} = \frac{6}{13} + \frac{4i}{13}\).
So the simplified form will be \(\frac{6}{13} + \frac{4}{13}i\).
This method of dealing with complex numbers is very handy in algebra and higher mathematics. It reduces complex division to a series of straightforward multiplication and addition steps, ensuring we end up with a standard form of a complex number. Don't forget, practice makes perfect!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Simplify. $$\frac{1}{2} \sqrt{36 a^{5} b c^{4}}-\frac{1}{2} \sqrt[3]{64 a^{4} b c^{6}}+\frac{1}{6} \sqrt{144 a^{3} b c^{6}}$$

Find the midpoint of each segment with the given endpoints. \(\left(-\frac{4}{5},-\frac{2}{3}\right)\) and \(\left(\frac{1}{8}, \frac{3}{4}\right)\)

Multiply and simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ \sqrt[3]{7 x} \sqrt[3]{3 x^{2}} $$

Simplify. $$ \frac{6}{1+\frac{3}{i}} $$

Multiply and simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ \sqrt{10} \sqrt{14} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.