/*! 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 77 Find each quotient. Write the an... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 77

Find each quotient. Write the answer in standard form \(a+b i .\) $$\frac{8}{-i}$$

Short Answer

Expert verified
The answer is \(0 + 8i\).

Step by step solution

01

- Eliminate the imaginary number from the denominator

To remove the imaginary number from the denominator, multiply both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of \(-i\) is \(i\). Hence, multiply numerator and denominator by \(i\).
02

- Multiply the numerator and denominator

Perform the multiplication: \(\frac{8}{-i} \times \frac{i}{i} = \frac{8i}{-i^2}\).
03

- Simplify the expression

Simplify the denominator first. Recall that \(i^2 = -1\), thus: \(-i^2 = -(-1) = 1\). So the expression becomes: \frac{8i}{1} = 8i\.
04

- Write in standard form

The standard form of a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers. For this problem, it simplifies directly to: \(0 + 8i\).

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.

Standard Form
When dealing with complex numbers, we often need to express our answers in a specific format called the standard form. The standard form of a complex number is written as \(a + bi\), where \(a\) and \(b\) are real numbers. The part \(a\) represents the real component, and \(bi\) represents the imaginary component.
For example, in the expression \(0 + 8i\), the real part is \(0\), and the imaginary part is \(8i\).
To write any complex number in standard form, simply separate its real and imaginary parts and ensure it fits the \(a + bi\) template. This form is essential because it allows us to easily compare and manipulate complex numbers.
Complex Conjugate
In the solution, we multiply by the complex conjugate to eliminate the imaginary number from the denominator. The complex conjugate of a complex number is found by changing the sign of its imaginary part. So, if you have a complex number \(a + bi\), its complex conjugate is \(a - bi\).
For example, the complex conjugate of \(-i\) is \(i\).
This trick is useful because multiplying a number by its conjugate results in a real number (since the imaginary parts cancel out).
Multiplying \([-i \times i]\) results in \(-i^2 = 1\), transforming the denominator into a real number, making it easier to simplify the expression.
Imaginary Number
An imaginary number is a number that, when squared, gives a negative result. In mathematics, the most common imaginary unit is represented by \(i\), which is defined as \(i^2 = -1\). Imaginary numbers are essential in defining complex numbers, which include a real part and an imaginary part.
In this exercise, \(i\) is used as the imaginary part, and it helps in simplifying expressions with complex numbers. When you see \(i\), remember that it essentially represents the square root of \(-1\).
This properties allows us to manipulate problems and express solutions that would otherwise be impossible with only real numbers.

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 each product. Write the answer in standard form. $$-5 i(4-3 i)^{2}$$

Explain why the following method of simplifying \(i^{-42}\) works. $$ i^{-42}=\frac{1}{i^{42}}=\frac{1}{\left(i^{2}\right)^{21}}=\frac{1}{(-1)^{21}}=\frac{1}{-1}=-1 $$

Find each quotient. Write the answer in standard form \(a+b i .\) $$ \frac{14+5 i}{3+2 i} $$

Find each quotient. Write the answer in standard form \(a+b i .\) $$\frac{2-i}{2+i}$$

Write each statement as an absolute value equation or inequality. \(k\) is within 0.0002 unit of \(10 .\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

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.