/*! 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 47 Find each sum or difference. Wri... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 47

Find each sum or difference. Write the answer in standard form. $$(2-5 i)-(3+4 i)-(-2+i)$$

Short Answer

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1 - 8i

Step by step solution

01

Distribute the Negative Sign

First, distribute the negative sign across the parentheses: }} This will change the expression to: (2-5i) - 3 - 4i - (-2 + i) which simplifies to: (2-5i) - 3 - 4i + 2 - i by flipping the signs inside the third parentheses.
02

Combine Like Terms

Now combine the real numbers (2, -3, and 2) and the imaginary numbers (-5i, -4i, and i): Real part: 2 - 3 + 2 = 1 Imaginary part: -5i - 4i + i = -8i
03

Write in Standard Form

Express the result as a complex number in standard form, which is a + bi: The result is 1 - 8i

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers
Complex numbers are a special type of number that combines a real part and an imaginary part. You can think of them as numbers that lie on a 2-dimensional plane, unlike the regular (real) numbers that lie on a 1-dimensional line.
The general form of a complex number is written as \(a + bi\), where:
  • \(a\) is the real part
  • \(b\) is the imaginary part
  • \(i\) is the imaginary unit, defined as \(i = \sqrt{-1}\)
For example, in the complex number \(3 + 2i\), 3 is the real part, and 2i is the imaginary part.
Standard Form
Writing complex numbers in standard form helps to clearly see both the real and imaginary parts. The standard form is \(a + bi\), and it's crucial for performing operations like addition and subtraction.
For instance, if you have two complex numbers like \(2 - 5i\) and \(3 + 4i\), writing them in standard form makes it easier to process steps to find the sum or difference.
Imaginary Numbers
Imaginary numbers represent a whole new dimension of numbers beyond the real number line. The basic unit is \(i = \sqrt{-1}\), an imaginary number that satisfies the equation \(i^2 = -1\).
When you perform arithmetic with imaginary numbers, treat the \(i\) part separately from the real part. For example, in the expression \(-5i - 4i + i\), you only combine the imaginary coefficients (-5, -4, and 1).
When adding or subtracting, think of \(i\) as a variable, like \(x\) in algebra, but you have to remember that \(i^2 = -1\).
Combining Like Terms
Combining like terms in complex numbers follows the same principle as in algebra. You group the real parts together and the imaginary parts together.
  • Real parts: These are the numbers without the imaginary unit \(i\). For example, in the expression \(2 - 3 + 2\), the real parts are 2, -3, and 2. When you add them, you get 1.
  • Imaginary parts: These are the numbers multiplied by \(i\). For instance, in \(-5i - 4i + i\), the coefficients are -5, -4, and 1. Adding them gives \(-8i\).
After combining, you write the result in standard form. For example, the final answer for the given exercise is \(1 - 8i\).

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Most popular questions from this chapter

Solve each problem. Recycling Bin Dimensions A recycling bin is in the shape of a rectangular box. Find the height of the box if its length is \(18 \mathrm{ft}\), its width is \(8 \mathrm{ft}\), and its surface area is \(496 \mathrm{ft}^{2}\). ( In the figure, \(h=\) height. Assume that the given surface area includes that of the top lid of the box.)

An astronaut on the moon throws a baseball upward. The astronaut is \(6 \mathrm{ft}, 6\) in. tall, and the initial velocity of the ball is \(30 \mathrm{ft}\) per sec. The height \(s\) of the ball in feet is given by the equation $$s=-2.7 t^{2}+30 t+6.5$$ where \(t\) is the number of seconds after the ball was thrown. (a) After how many seconds is the ball \(12 \mathrm{ft}\) above the moon's surface? Round to the nearest hundredth. (b) How many seconds will it take for the ball to return to the surface? Round to the nearest hundredth.

Explain how to determine whether to use a parenthesis or a square bracket when graphing the solution set of a linear inequality.

Solve each rational inequality. Write each solution set in interval notation.4 $$9\frac{(5 x-3)^{3}}{(25-8 x)^{2}} \leq 0$$ $$\frac{(2 x-3)(3 x+8)}{(x-6)^{3}} \geq 0 \quad\( 92. \)\frac{(9 x-11)(2 x+7)}{(3 x-8)^{3}}>0$$.

Classroom Ventilation According to the American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (ASHRAE), a nonsmoking classroom should have a ventilation rate of \(15 \mathrm{ft}^{3}\) per min for each person in the room. (a) Write an equation that models the total ventilation \(V\) (in cubic feet per hour) necessary for a classroom with \(x\) students. (b) A common unit of ventilation is air change per hour (ach). 1 ach is equivalent to exchanging all of the air in a room every hour. If \(x\) students are in a classroom having volume \(15,000 \mathrm{ft}^{3},\) determine how many air exchanges per hour \((A)\) are necessary to keep the room properly ventilated. (c) Find the necessary number of ach ( \(A\) ) if the classroom has 40 students in it. (d) In areas like bars and lounges that allow smoking, the ventilation rate should be increased to \(50 \mathrm{ft}^{3}\) per min per person. Compared to classrooms, ventilation should be increased by what factor in heavy smoking areas?
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