/*! 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 46 Write each quotient in the form ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 46

Write each quotient in the form \(a+\) bi. $$ \frac{1}{5+2 i} $$

Short Answer

Expert verified
\( \frac{1}{5+2i} = \frac{5}{29} - \frac{2i}{29} \)

Step by step solution

01

Rationalize the Denominator

To write the fraction in the form of \(a + bi\), multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \(5 + 2i\) is \(5 - 2i\). Multiply both the numerator and denominator by \(5 - 2i\): \[ \frac{1}{5 + 2i} \times \frac{5 - 2i}{5 - 2i} = \frac{5 - 2i}{(5 + 2i)(5 - 2i)} \]
02

Simplify the Denominator Using Difference of Squares

Simplify the denominator by using the difference of squares formula: \[ (5 + 2i)(5 - 2i) = 25 - (2i)^2 \] Since \(i^2\) is \(-1\), \[ = 25 - 4(-1) = 25 + 4 = 29 \]
03

Simplify the Fraction

Now, place the simplified denominator value in the fraction: \[ \frac{5 - 2i}{29} = \frac{5}{29} - \frac{2i}{29} \]
04

Write in the Form a + bi

Write the final answer using the form \(a + bi\). Here, \[ a = \frac{5}{29} \] and \[ b = \frac{-2}{29} \] Therefore, \[ \frac{1}{5 + 2i} = \frac{5}{29} - \frac{2i}{29} \]

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 the Denominator
Rationalizing the denominator is a method used to simplify fractions that contain complex numbers. When a denominator has a complex number, we take its conjugate and use it to eliminate the imaginary unit from the denominator.
The conjugate of a complex number involves changing the sign between the real and imaginary parts. For example, the conjugate of \(5 + 2i\) is \(5 - 2i\).
By multiplying both the numerator and the denominator by this conjugate, we simplify the expression:
\[ \frac{1}{5 + 2i} \times \frac{5 - 2i}{5 - 2i} = \frac{5 - 2i}{(5 + 2i)(5 - 2i)} \] This operation helps to remove the imaginary unit from the denominator.
Difference of Squares
The difference of squares formula is a useful algebraic identity for simplifying expressions involving squares. The formula states:
\[ (a + b)(a - b) = a^2 - b^2 \]
When dealing with complex numbers, this formula is used to simplify the product of a complex number and its conjugate. For example, consider \(5 + 2i\) and \(5 - 2i\). Using the difference of squares formula, we get:
\[ (5 + 2i)(5 - 2i) = 5^2 - (2i)^2 \] Since \(i^2 = -1\), the expression becomes:
\[ 25 - (2i)^2 = 25 - 4(-1) = 25 + 4 = 29 \] This result shows that the imaginary parts cancel out, leaving a real number denominator.
Imaginary Unit
The imaginary unit, denoted by \(i\), is defined as the square root of -1. It is a fundamental concept in complex numbers. Key properties of the imaginary unit include:
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)

When working with complex numbers, it's important to remember these properties as they help in simplifying complex expressions. In the given exercise, the imaginary part \(2i\) and its square \( (2i)^2 \) are calculated as follows:
\[ (2i)^2 = 4i^2 = 4(-1) = -4 \]
These steps demonstrate how the imaginary unit transforms under multiplication and helps in rationalizing and simplifying the fractions in complex number operations.

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

For each rectangular equation, write an equivalent polar equation. $$ y=-3 x+5 $$

Graph each pair of polar equations on the same screen of your calculator and use the trace feature to estimate the polar coordinates of all points of intersection of the curves. Check your calculator manual to see how to graph polar equations. $$ r=1, r=2 \sin 3 \theta $$

Write each expression in the form \(a+\) bi where \(a\) and \(b\) are real numbers. $$ \frac{6+\sqrt{-8}}{2} $$

Write \(4-4 i\) in trigonometric form, using degree measure for the argument.

Sketch the graph of each pair of parametric equations. $$ x=t-3, y=t^{2}, \text { for } t \text { in }(-\infty, \infty) $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

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.