/*! 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 71 Find the following quotients. Wr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 71

Find the following quotients. Write all answers in standard form for complex numbers. $$\frac{5-2 i}{i}$$

Short Answer

Expert verified
The quotient is \( 2 + 5i \).

Step by step solution

01

Set up the expression

Given the expression \( \frac{5 - 2i}{i} \), our goal is to express this quotient in standard form \( a + bi \), where \( a \) and \( b \) are real numbers.
02

Multiply by the conjugate of the denominator

The conjugate of \( i \) is \( -i \). Multiply both the numerator and denominator by \( -i \) to eliminate the imaginary unit in the denominator: \[ \frac{(5 - 2i)(-i)}{i(-i)} \].
03

Simplify the denominator

Simplify the denominator using the property that \( i^2 = -1 \): \( i(-i) = i^2 = -1 \). Thus, the denominator becomes \(-1\).
04

Distribute and simplify the numerator

Distribute \( -i \) in the numerator:\( (5 - 2i)(-i) = 5(-i) - 2i(-i) = -5i + 2i^2 \).Since \( i^2 = -1 \), this becomes \(-5i + 2(-1) = -5i - 2 \).
05

Simplify the entire expression

The expression now is \( \frac{-5i - 2}{-1} \). Simplify this by dividing each term by \(-1\):\( \frac{-5i}{-1} + \frac{-2}{-1} = 5i + 2 \).So the result in standard form is \( 2 + 5i \).

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 Numbers Division
Dividing complex numbers might seem challenging at first, but it follows a logical process. When dividing a complex number like \( \frac{5 - 2i}{i} \), the main goal is to remove the imaginary unit \( i \) from the denominator. This process requires multiplication by the conjugate.
  • **Identify the denominator**: In this exercise, the denominator is simply \( i \).
  • **Find the conjugate**: The conjugate of \( i \) is \( -i \), because multiplying \( i \) by \(-i\) results in \( i^2 = -1 \), removing the imaginary unit.
  • **Multiply the numerator and the denominator by the conjugate**: Perform the multiplication to keep the expression equivalent but without \( i \) in the denominator.
This method ensures both the numerator and denominator have been transformed to help achieve a real number result or a simplified complex number expression.
Standard Form of Complex Numbers
After handling the division, the next step is to ensure the result is in standard form, which is \( a + bi \).
  • **Understand the form**: For complex numbers, standard form means having a real part \( a \) and an imaginary part \( bi \).
  • **Simplifying the expression**: If the division or any other operation results in non-standard form, you'll need to collect terms.
For instance, in our final result \( 2 + 5i \), the real part is \( 2 \) and the imaginary part is \( 5i \). This order and structure help to clearly identify the respective components and correctly classify the number as a complex number in standard form.
Simplifying Complex Expressions
Simplifying complex expressions is crucial for clearer results and easier understanding of complex numbers.
  • **Distribute accurately**: Make sure each part of the binomial gets multiplied correctly when expanding. For example, with \( (5 - 2i)(-i) \), each term must interact with \(-i\).
  • **Use \( i^2 = -1 \) appropriately**: This property turns imaginary terms into real values when combined in pairs.
  • **Combine like terms**: After distribution and substitution with \( i^2 = -1 \), sum up the real terms separately from the imaginary terms.
Following these simplification steps can transform a seemingly complex expression into an easily manageable format, making it much simpler to interpret and utilize in calculations.

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

Find \(\theta\) between \(0^{\circ}\) and \(360^{\circ}\) based on the given information. \(\sin \theta=-\frac{1}{2}\) and \(\theta\) terminates in QIII

Let \(z_{1}=2+3 i, z_{2}=2-3 i\), and \(z_{3}=4+5 i\), and find $$z_{1} z_{2}$$

Let \(z_{1}=2+3 i, z_{2}=2-3 i\), and \(z_{3}=4+5 i\), and find $$z_{3}\left(z_{1}-z_{2}\right)$$

Find the following quotients. Write all answers in standard form for complex numbers. $$\frac{2 i}{3+i}$$

Find the following products. $$(7+2 i)(7-2 i)$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

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.