/*! 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 50 Evaluate the expression with a c... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 50

Evaluate the expression with a calculator. $$ (-8.05-4.67 i)+(3.5+5.37) $$

Short Answer

Expert verified
-4.55 + 0.70i

Step by step solution

01

Identify Real and Imaginary Parts

Observe that the expression is the sum of two complex numbers: (-8.05 - 4.67i) and (3.5 + 5.37). Notice the first term has a real part of -8.05 and an imaginary part of -4.67i. The second term has a real part of 3.5 and an imaginary part of 5.37i.
02

Add the Real Parts

Add the real parts of the complex numbers: -8.05 (from the first number) and +3.5 (from the second number). Calculate: -8.05 + 3.5 = -4.55.
03

Add the Imaginary Parts

Add the imaginary parts of the complex numbers: -4.67i (from the first number) and +5.37i (from the second number). Calculate: -4.67 + 5.37 = 0.70.
04

Combine the Results

Combine the sums from Steps 2 and 3 to form the resulting complex number: -4.55 + 0.70i.

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.

Real and Imaginary Parts
Complex numbers consist of a real part and an imaginary part. A complex number is typically represented in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part with \(b\) being a real number and \(i\) representing the square root of -1.

For example, consider the complex number \(-8.05 - 4.67i\). Here, the real part is \(-8.05\) and the imaginary part is \(-4.67i\). Similarly, for the complex number \(3.5 + 5.37i\), the real part is \(3.5\) and the imaginary part is \(5.37i\).

Identifying these parts is crucial. It helps you perform operations on complex numbers, such as addition, fairly straightforwardly.
Addition of Complex Numbers
Adding complex numbers is much like combining like terms in algebra. For any two complex numbers, say \(a + bi\) and \(c + di\), you add the real parts together and the imaginary parts together separately.

Consider the expression \((-8.05 - 4.67i) + (3.5 + 5.37i)\).
  • First, add the real parts: \(-8.05 + 3.5 = -4.55\).
  • Next, add the imaginary parts: \(-4.67 + 5.37 = 0.70\).
By adding these separately, you ensure each part retains its respective value, resulting in the final complex number \(-4.55 + 0.70i\).

This operation shows how complex numbers can be manipulated through straightforward arithmetic, simplifying otherwise complicated equations.
Mathematical Expressions Evaluation
Evaluating mathematical expressions involving complex numbers often requires breaking them down into simpler parts, like separating real and imaginary components.

In our example, we used a calculator to streamline the calculation steps for the expression \((-8.05 - 4.67i) + (3.5 + 5.37i)\). By directly adding the individual components, both real and imaginary, we simplify the problem to basic addition.
  • First, simplify the real parts.
  • Then simplify the imaginary parts.
  • Finally, combine these results to obtain the full evaluated expression.
Using tools like calculators can further ensure accuracy, especially when dealing with decimals or more complex expressions. This method breaks down intimidating expressions into manageable tasks, making complex math much less daunting.

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

As the altitude increases, air becomes thinner, or less dense. An approximation of the density of air at an altitude of \(x\) meters above sea level is given by \(d(x)=\left(3.32 \times 10^{-9}\right) x^{2}-\left(1.14 \times 10^{-4}\right) x+1.22\) The output is the density of air in kilograms per cubic meter. The domain of \(d\) is \(0 \leq x \leq 10,000\). (a) Denver is sometimes referred to as the mile-high city. Compare the density of air at sea level and in Denver. (Hint: 1 ft \(\approx 0.305\) m.) (b) Determine the altitudes where the density is greater than 1 kilogram per cubic meter.

Use transformations to sketch a graph of \(f\). \(f(x)=\sqrt{x-3}+2\)

Use transformations of graphs to model the table of data with the formula \(f(x)=a(x-h)^{2}+k .\) (Answers may vary.) Number of titles released for DVD rentals $$ \begin{array}{llllll} \text { Year } & 1998 & 1999 & 2000 & 2001 & 2002 \\ \text { Titles } & 2049 & 4787 & 8723 & 14,321 & 21,260 \end{array} $$

Two functions, \(f\) and \(g,\) are related by the given equation. Use the numerical representation of \(f\) to make a numerical representation of \(\mathbf{g}\). \(g(x)=f(-x)+1\) $$ \begin{array}{rrrrrr} x & -2 & -1 & 0 & 1 & 2 \\ f(x) & 11 & 8 & 5 & 2 & -1 \end{array} $$

Two functions, \(f\) and \(g,\) are related by the given equation. Use the numerical representation of \(f\) to make a numerical representation of \(\mathbf{g}\). \(g(x)=f(x-3)+5\) $$\begin{array}{rrrrrr}x & -3 & 0 & 3 & 6 & 9 \\\\\hline f(x) & 3 & 8 & 15 & 27 & 31\end{array}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Calculus

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.