/*! 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 29 Write the expression in standard... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 29

Write the expression in standard form. $$ (2)(2+4 i) $$

Short Answer

Expert verified
4 + 8i

Step by step solution

01

Distribute the real number

To write \((2)(2+4i)\) in standard form, start by applying the distributive property. This means we will multiply 2 by each part inside the parentheses.Multiply 2 by 2:\[2 \times 2 = 4\]Multiply 2 by \(4i\):\[2 \times 4i = 8i\]
02

Combine and simplify

Now that each term has been distributed, combine the results from Step 1. The expression \(4 + 8i\) is already in its simplest form.The real part is 4, and the imaginary part is \(8i\).

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 in Complex Numbers
Complex numbers are numbers of the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. The standard form of a complex number highlights both its real and imaginary components, which makes it easier to perform arithmetic operations and understand the properties of the number.
  • The real part, \( a \), represents the horizontal component on a complex plane.
  • The imaginary part, \( bi \), represents the vertical component, where \( b \) is a real number, and \( i \) indicates the imaginary unit.
In our exercise, after simplification, we obtain \( 4 + 8i \) in standard form. Here, 4 is the real part, and \( 8i \) is the imaginary part. This format helps in clearly distinguishing the components of the complex number, and such clarity is crucial for both manual calculations and computer-generated solutions in mathematics.
Understanding the Distributive Property
The distributive property is a fundamental concept in mathematics, especially when dealing with algebraic expressions. It allows us to multiply a single term by each term within a set of parentheses independently in a structured manner. The property is expressed as:\[ a(b + c) = ab + ac \]Using the distributive property makes computations straightforward and minimizes errors. In our example, we applied the distributive property to the expression \( (2)(2 + 4i) \):
  • First, multiply 2 (the outside term) by 2, producing 4.
  • Next, multiply 2 by \( 4i \), resulting in \( 8i \).
This methodical distribution yields the expression \( 4 + 8i \) in standard form. Recognizing and utilizing the distributive property is essential for simplifying complex expressions accurately.
The Imaginary Unit and Its Importance
The imaginary unit, denoted as \( i \), is a unique symbol in mathematics with a special property: \( i^2 = -1 \). This concept extends the number system to include values that are not on the real number line.
  • While real numbers have their place on a linear number system, complex numbers reside on a plane formed by the real axis and the imaginary axis.
  • The imaginary unit helps in solving equations that have no real solutions, offering a complete set of solutions for polynomial equations.
In our exercise, the term \( 4i \) becomes \( 8i \) after applying the distributive property, showcasing how \( i \) operates within complex numbers. Understanding the imaginary unit is crucial when working in fields like electrical engineering, control systems, and anywhere complex numbers are used for modeling signals and waves.

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

If the graph of \(y=f(x)\) undergoes a vertical stretch or shrink to become the graph of \(y=g(x),\) do these two graphs have the same \(x\) -intercepts? \(y\) -intercepts? Explain your answers.

The points \((-12,6),(0,8),\) and \((8,-4)\) lie on the graph of \(y=f(x)\). Determine three points that lie on the graph of \(y=g(x)\). \(g(x)=f(-2 x)\)

The table lists the velocity and distance raveled by a falling object for various elapsed times. $$ \begin{array}{|rcccccc|} \hline \text { Time (sec) } & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Velocity (fl/sec) } & 0 & 32 & 64 & 96 & 128 & 160 \\ \hline \text { Distance (ft) } & 0 & 16 & 64 & 144 & 256 & 400 \\ \hline \end{array} $$ (a) Make a scatterplot of the ordered pairs determined by (time, velocity) and (time, distance) in the same vicwing rectangle \([-1,6,1]\) by \([-10,450,50]\) (b) Find a function \(v\) that models the velocity. (c) The distance is modeled by \(d(x)=a x^{2} .\) Find \(a\) (d) Find the time when the distance is 200 feet. Find the velocity at this time.

The stopping distance \(D\) in feet for a car traveling at \(x\) miles per hour on wet level pavement can be estimated by \(D(x)=\frac{1}{9} x^{2}+\frac{11}{3} x\). If a driver can see only 300 feet ahead on a curve, find a safe speed limit.

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

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

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.