Problem 35 In $35-43,$ write each number ... [FREE SOLUTION]

Chapter 5: Problem 35

In $35-43,$ write each number in simplest form. $$ 3 i+2 i^{3} $$

Short Answer

Expert verified

The expression simplifies to $ i $.

Step by step solution

Simplify powers of i

We know the powers of the imaginary unit, i, follow a pattern: $ i = i $, $ i^2 = -1 $, $ i^3 = -i $, $ i^4 = 1 $, and then the pattern repeats. Here, we have $ i^3 = -i $.

Substitute and simplify

Substitute $ i^3 $ with $ -i $ in the expression $ 3i + 2i^3 $, yielding $ 3i + 2(-i) $.

Combine like terms

Combine the terms: $ 3i + 2(-i) = 3i - 2i = i $.

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.

Imaginary Unit

The concept of the imaginary unit, commonly denoted as $ i $, is a fundamental idea in complex numbers. Originally, mathematicians faced challenges when trying to find the square root of negative numbers. To solve this, they introduced $ i $, a number which satisfies the equation $ i^2 = -1 $. This means:\
\[\sqrt{-1} = i\]
With this definition, you're able to handle operations involving square roots of negative numbers smoothly.

It allows for the extension of real numbers to complex numbers, which are numbers of the form $a + bi$, where $a$ and $b$ are real numbers.
This concept turns complex numbers into a powerful tool in various fields such as engineering, physics, and computer science.

Powers of i

Understanding the powers of the imaginary unit $ i $ is essential when working with complex numbers. The powers of $ i $ follow a cyclical pattern:

$ i^1 = i $
$ i^2 = -1 $
$ i^3 = -i $
$ i^4 = 1 $

After $ i^4 $, the pattern repeats itself. So $ i^5 = i $, $ i^6 = -1 $, and so on.

To determine any power of $ i $, divide the exponent by 4 and use the remainder to find where it fits in the cycle:

If the remainder is 0, the power of $ i $ is 1.
If the remainder is 1, the power of $ i $ is $ i $.
If the remainder is 2, the power of $ i $ is -1.
If the remainder is 3, the power of $ i $ is -i.

Simplifying Expressions

Simplifying expressions involving complex numbers often revolves around handling the imaginary unit and its powers correctly. Here's a simple formula to consider: when given an expression like $ 3i + 2i^3 $, you should follow these steps to simplify:

Identify and substitute powers of $ i $: Recognize that $ i^3 = -i $ from the pattern discussed previously.
Replace $ i^3 $ with $-i$ to get $ 3i + 2(-i) $.
Combine like terms: $ 3i + 2(-i) = 3i - 2i = i $.

Utilizing these steps ensures you consistently simplify complex expressions accurately and efficiently, avoiding common pitfalls when dealing with imaginary numbers.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

In \(35-43,\) write each number in simplest form. $$ 3 i+2 i^{3} $$

Short Answer

Step by step solution

Simplify powers of i

Substitute and simplify

Combine like terms

Key Concepts

Imaginary Unit

Powers of i

Simplifying Expressions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Logic and Functions

Decision Maths

Theoretical and Mathematical Physics

Applied Mathematics

Geometry

Study anywhere. Anytime. Across all devices.