/*! 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 48 What is the complex conjugate of... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 48

What is the complex conjugate of \(2+3 i ?\) What happens when you multiply this complex number by its complex conjugate?

Short Answer

Expert verified
The complex conjugate of \(2+3i\) is \(2-3i\), and multiplying this complex number by its conjugate results in -5.

Step by step solution

01

Finding the Complex Conjugate

A complex number can be represented in the form \(a+bi\), where \(a\) and \(b\) are real numbers and \(i\) is the square root of -1. The complex conjugate of \(a+bi\) is \(a-bi\). Thus, to find the complex conjugate of \(2+3i\), we simply need to change the sign of the imaginary part.
02

Calculation

Changing the sign of the imaginary part in \(2+3i\) leads to \(2-3i\). This is the complex conjugate.
03

Multiplying the Complex Number and its Conjugate

To multiply the complex number \(2+3i\) and its conjugate \(2-3i\), we simply apply the rule of 'FOIL' as with binomials. Thus, we calculate \( (2+3i)*(2-3i) = (2*2) + (2*-3i) +(3i*2)+ (3i*-3i)\).
04

Simplifying the Result

Simplifying the expression from step 3 results in \(4 - 6i + 6i -9 = 4 -9 = -5\) because \(i^2 = -1\).

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!

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

Which one of the following is true? a. Some irrational numbers are not complex numbers. b. \((3+7 i)(3-7 i)\) is an imaginary number. c. \(\frac{7+3 i}{5+3 i}=\frac{7}{5}\) d. In the complex number system, \(x^{2}+y^{2}\) (the sum of two squares) can be factored as \((x+y i)(x-y i)\)

Solve each rational inequality in Exercises \(29-48,\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x+5}{x-2}>0 $$

In Exercises \(73-74\), use the method for solving quadratic inequalities to solve each higher-order polynomial inequality. $$ x^{3}+2 x^{2}-x-2 \geq 0 $$

In Exercises \(29-44,\) perform the indicated operations and write the result in standard form. $$(3 \sqrt{-7})(2 \sqrt{-8})$$

In Exercises \(9-20,\) find each product and write the result in standard form. $$(8-4 i)(-3+9 i)$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

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.