/*! 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 70 In Exercises \(63-84,\) divide a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 70

In Exercises \(63-84,\) divide and simplify to the form \(a+b i\) $$\frac{5 i}{2-3 i}$$

Short Answer

Expert verified
The simplified form of the given expression is \(-15/13 + 10/13 * i\).

Step by step solution

01

Calculate the complex conjugate

Find the complex conjugate of the denominator \(2 - 3i\). The complex conjugate is \(2 + 3i\). This is done by changing the sign of the imaginary part.
02

Multiply the numerator and the denominator by the complex conjugate

Now, we multiply the numerator and the denominator by \(2 + 3i\). This gives: \[\frac{5i(2 + 3i)}{(2 - 3i)(2 + 3i)}.\]
03

Simplify the expression

First, distribute \(5i\) through the numerator: \[5i * 2 + 5i * 3i = 10i - 15,\] because \(i^2 = -1\). Next, multiply the numbers in the denominator: \[(2 - 3i)(2 + 3i) = 4 + 9 = 13.\] So the simplified form is \(\frac{10i - 15}{13}\). In the form \(a + bi\), \(a = -15/13\) and \(b = 10/13\). This means our final answer is \(-15/13 + 10/13 * i\).

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.

Complex Conjugate
Understanding the concept of a complex conjugate is essential when dealing with complex numbers, especially in the context of division. Imagine you have a complex number, such as the denominator in our example exercise, which is defined as the sum of a real part and an imaginary part, written in the form of \( a + bi \). Its complex conjugate is simply given by changing the sign of the imaginary term, resulting in \( a - bi \).

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

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. After squaring both sides of a radical equation, the only solution that I obtained was extraneous, so \(\varnothing\) must be the solution set of the original equation.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \(-\sqrt{x}=9\) has no solution.

Let \(f(x)=x^{2}+4 x-2 .\) Find \(f(-2+\sqrt{6})\)

Exercises \(147-149\) will help you prepare for the material covered in the next section. Solve: \(26-11 x=16-8 x+x^{2}\)

In Exercises \(75-92,\) rationalize each denominator. Simplify, if possible. $$\frac{2 \sqrt{x}+\sqrt{y}}{\sqrt{y}-2 \sqrt{x}}$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

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.