/*! 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 80 Simplify. \(i^{43}\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 80

Simplify. \(i^{43}\)

Short Answer

Expert verified
\(i^{43} = -i\)

Step by step solution

01

Understanding Powers of i

The imaginary unit i follows a cyclical pattern with its powers: \(i = \sqrt{-1}\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). These results repeat every four powers.
02

Determine the Position within the Cycle

To simplify \(i^{43}\), find the remainder when 43 is divided by 4 (since the pattern repeats every 4 powers). \(43 \div 4 = 10\) with a remainder of 3. Thus, \(i^{43}\) is equivalent to \(i^{3}\).
03

Simplify Using the Pattern

Using the known cyclic pattern, \(i^3 = -i\). Therefore, \(i^{43} = -i\).

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.

Powers of i
Imaginary numbers are a fascinating part of mathematics, especially the imaginary unit, denoted as i. The imaginary unit is defined as the square root of -1, that is, \(i = \sqrt{-1}\). This seemingly simple definition leads to a sequence of powers with interesting properties.

The powers of i follow a repetitive or cyclic pattern:
  • \(i = \sqrt{-1}\)
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)
Notice that after the fourth power, the sequence starts again:

\[ \begin{align*} i^5 &= i \ i^6 &= -1 \ i^7 &= -i \ i^8 &= 1 \end{align*} \] Understanding this cycle is essential because it means we can simplify any power of i by recognizing this repeating pattern.
Cyclic Pattern
As mentioned earlier, the powers of i repeat every four powers, creating a cyclic pattern. This pattern greatly simplifies the process of dealing with higher powers of i.

For example, let's consider a large power such as \(i^{43}\). Without the cyclic pattern, calculating this directly would be tedious. However, recognizing that every fourth power is the same as one of the first four powers simplifies this task immensely.

The cyclic pattern can be summarized as follows:
  • \(i^{4k} = 1\)
  • \(i^{4k+1} = i\)
  • \(i^{4k+2} = -1\)
  • \(i^{4k+3} = -i\)
Here, k is a whole number. This repetition every four powers is what makes calculations with i manageable and simple.
Remainder
To use the cyclic pattern efficiently, particularly with large exponents, we can utilize the concept of the remainder when dividing by 4. Given an exponent, finding how many complete cycles of 4 fit in, and what remains, can provide the equivalent power in the first cycle.

Let's take \(i^{43}\) as an example:
  • First, divide the exponent by 4: \(43 \div 4 = 10\) with a remainder of 3.
  • This tells us \(i^{43}\) is equivalent to \(i^3\) because 43 is 10 full cycles of 4 plus an extra 3.
Using the cyclic pattern where \(i^3 = -i\), we can easily conclude \(i^{43} = -i\).

This remainder approach saves time and makes simplifying powers of i straightforward. Whenever you face such an exponent, just find the remainder when dividing by 4, and refer back to the initial cycle of powers of i.

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

The decay rate of krypton- 85 is \(6.3 \%\) per year. What is its half-life?

The number of people who have heard a rumor increases exponentially. If each person who hears a rumor repeats it to two people per day, and if 20 people start the rumor, the number of people \(N\) who have heard the rumor after \(t\) days is given by $$ N(t)=20(3)^{t} $$ a) After what amount of time will 1000 people have heard the rumor? b) What is the doubling time for the number of people who have heard the rumor?

Simplify. \(\sqrt[3]{40 a^{5} b^{12}}\)

Classify each of the following statements as either true or false. To solve an exponential equation, we can take the common logarithm of both sides of the equation.

The moose population in New York is growing exponentially. The number of moose in the state \(t\) years after 1997 can be approximated by $$ M(t)=50(1.25)^{t} $$ a) Estimate the number of moose in New York in \(1997,\) in \(2012,\) and \(\operatorname{in} 2020\) b) Graph the function. (IMAGE CANT COPY)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

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.