/*! 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 74 Find each quotient. $$ \frac... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 74

Find each quotient. $$ \frac{5+i}{-i} $$

Short Answer

Expert verified
The quotient is \-1 + 5i\.

Step by step solution

01

Identify the Expression

We are given the complex fraction \(\frac{5+i}{-i}\). Our goal is to simplify this expression.
02

Multiply by the Conjugate

To eliminate the imaginary denominator, multiply both numerator and denominator by the conjugate of the denominator. The conjugate of \(-i\) is \(-i\). Thus, multiply and divide by \(-i\): \(\frac{(5+i)(-i)}{(-i)(-i)}\).
03

Simplify the Denominator

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

Simplify the Numerator

Expand and simplify the numerator: \((5+i)(-i) = 5(-i) + i(-i) = -5i - i^2\). Since \i^2 = -1\, the numerator becomes: \-5i - (-1) = -5i + 1= 1 - 5i\.
05

Form the Final Simplified Expression

Combine the simplified numerator and denominator: \(\frac{1-5i}{-1}\). This is equivalent to dividing each term in the numerator by the denominator: \(-1 + 5i\).

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 Fraction
A complex fraction involves complex numbers in its numerator, denominator, or both. These numbers have a real part and an imaginary part. For instance, the fraction given in the exercise \(\frac{5+i}{-i}\) includes both a real number, 5, and an imaginary number, \(i\). Simplifying complex fractions often requires the use of the conjugate to eliminate imaginary units from the denominator. This ensures the expression is easier to work with and understand.
Conjugate
The conjugate of a complex number is found by changing the sign of its imaginary part. For example, the conjugate of \(a + bi\) is \(a - bi\). In our problem, the conjugate of \(-i\) is still \(-i\) because it has no real part. Using the conjugate is crucial for simplifying complex fractions as it helps to eliminate the imaginary part from the denominator.

Multiplying the numerator and the denominator by the conjugate helps achieve this, transforming complex expressions into a more easily manageable form.
Imaginary Unit
The imaginary unit is represented by the symbol \(i\), where \(i^2 = -1\). This is a fundamental concept in complex numbers. In the given exercise, \(-i\) is the imaginary unit and is in the denominator of the complex fraction. To simplify our complex fraction, recognizing that \(i^2 = -1\) allows us to properly simplify expressions involving imaginary units.

When dealing with multiplications involving the imaginary unit, the relationships \i^2 = -1\ and \-i^2 = -1\ make it easier to eliminate imaginary numbers from the denominator, giving us a clearer expression.

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

Solve each equation. $$ 4 x-7=0 $$

Rationalize each denominator. Assume that all radicals represent real numbers and no denominators are 0. $$ \frac{3}{\sqrt{x+y}} $$

Find the distance between each pair of points. $$ (8,13) \text { and }(2,5) $$

Combine like terms. $$ 13 x^{4}-12 x^{3}+9 x^{4}+2 x^{3} $$

Write with rational exponents, and then apply the properties of exponents. Assume that all radicands represent posititive real mumbers. Give answers in exponential form. See Example 6. Suppose someone claims that \(\sqrt[n]{a^{n}+b^{n}}\) must equal \(a+b,\) since, when \(a=1\) and \(b=0\) a true statement results: $$ \sqrt[n]{a^{n}+b^{n}}=\sqrt[n]{1^{n}+0^{n}}=\sqrt[n]{1^{n}}=1=1+0=a+b $$ Explain why this is faulty reasoning.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Mechanics 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.