/*! 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 30 Write the expression in standard... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 30

Write the expression in standard form. $$ (-5)(-7+3 i) $$

Short Answer

Expert verified
The expression in standard form is \(35 - 15i\).

Step by step solution

01

Apply the Distributive Property

To distribute (-5) across the complex number (-7 + 3i), apply the distributive property: (-5) imes (-7) + (-5) imes (3i).
02

Calculate Each Term

Calculate the product of each distributed term: 1. (-5) imes (-7) = 35 2. (-5) imes (3i) = -15i. This gives us the expression: 35 - 15i.
03

Write in Standard Form

The standard form for a complex number is a + bi, where a and b are real numbers. In this case, 35 - 15i is already in standard form: 35 + (-15)i, where a = 35 and b = -15.

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.

Distributive Property
The distributive property is a fundamental concept in algebra that makes simplifying expressions easier. When you apply the distributive property, you multiply a single term by each term inside a bracket. This property is a useful tool when dealing with both real numbers and complex numbers.
  • A complex number is often written in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit.
  • The distributive property allows you to expand expressions by distributing a multiplying factor across the terms within a parenthesis.
To illustrate, let's consider \((-5)(-7 + 3i)\). By using the distributive property, you multiply \(-5\) by each of \(-7\) and \(3i\), resulting in two new terms. This step is crucial to simplifying and eventually finding the solution for expressions that include complex numbers.
Standard Form
When working with complex numbers, it is important to express the final result in standard form. The standard form of a complex number is written as \(a + bi\), where \(a\) and \(b\) are real numbers.
  • The real part, \(a\), represents the real number in the expression.
  • The imaginary part, \(bi\), includes both the coefficient \(b\) and the imaginary unit \(i\).
This format is handy because it clearly shows the real and imaginary components of the number. Using our earlier example, after distributing and simplifying \((-5)(-7) + (-5)(3i)\), we obtain \(35 - 15i\). Observing this expression, we see that it perfectly adheres to the standard form of a complex number where \(a = 35\) and \(b = -15\). It鈥檚 clear and organized, making it easier to work with complex numbers in future calculations.
Imaginary Unit
The imaginary unit, denoted as \(i\), is a special number that doesn't quite fit into the world of real numbers. It is defined as \(i = \sqrt{-1}\). This unique property of \(i\) allows for the representation and operation of complex numbers.
  • When squared, \(i^2\) equals \(-1\). This can lead to interesting results in calculations and is a foundational factor in complex numbers.
  • Complex numbers are built on the combination of real numbers and multiples of \(i\), making them versatile for solving equations involving square roots of negative numbers.
In the expression \(35 - 15i\), \(i\) signifies the imaginary part, with \(-15\) being the coefficient of the imaginary unit. Understanding \(i\) is key to grasping complex number operations, as it allows you to work with expressions that would otherwise lack real solutions.

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

(Refer to Example 12.) A company charges \(\$ 20\) to make one monogrammed shirt, but reduces this cost by \(\$ 0.10\) per shirt for each additional shirt ordered up to 100 shirts. If the cost of an order is \(\$ 989,\) how many shirts were ordered?

Braking distance for cars on level pavement can be approximated by \(D(x)=\frac{x^{2}}{30 k^{*}}\) The input \(x\) is the car's velocity in miles per hour and the output \(D(x)\) is the braking distance in feet. The positive constant \(k\) is a measure of the traction of the tires. Small values of \(k\) indicate a slippery road or worn tires. (Source: L. Haefner, Introduction to Transportation Systems.) (a) Let \(k=0.3 .\) Evaluate \(D(60)\) and interpret the result. (b) If \(k=0.25,\) find the velocity \(x\) that corresponds to a braking distance of 300 feet.

The number \(N\) of women in millions who were gainfully employed in the work force in selected years is shown in the table. $$ \begin{array}{|rccccccc} \hline \text { Year } & 1900 & 1910 & 1920 & 1930 & 1940 & 1950 \\ \hline N & 5.3 & 7.4 & 8.6 & 10.8 & 12.8 & 18.4 \\ \hline \text { Year } & 1960 & 1970 & 1980 & 1990 & 2000 & 2010 \\ \hline N & 23.2 & 31.5 & 45.5 & 56.6 & 65.6 & 74.8 \end{array} $$ (a) Use regression to find a quadratic function \(f\) that models the data. Support your result graphically. (b) Predict the number of women in the labor force in 2020

Solve. Write answers in standard form. $$ 2 x^{2}+3=1-x $$

Use transformations to sketch a graph of \(f\). \(f(x)=|2 x|\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

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.