/*! 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 6 Decide whether each expression i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 6

Decide whether each expression is equal to \(1,-1, i, o r-i\) $$ (-i)^{2} $$

Short Answer

Expert verified
The expression \((-i)^2\) is equal to -1.

Step by step solution

01

Understand Powers of Complex Numbers

Complex numbers often include the imaginary unit, represented as \(i\), where \(i^2 = -1\). It's important to understand the powers of the imaginary unit: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\), and then the cycle repeats.
02

Calculate \((-i)^2\)

To calculate \((-i)^2\), recall that \((-i) = -1 \times i\). Squaring it gives: \[ (-i)^{2} = (-1 \times i)^2 = (-1)^2 \times i^2 \]
03

Simplify the Expression

Simplify further using the rule for exponents and the property of imaginary numbers: \[ (-1)^2 = 1 \quad \text{and} \quad i^2 = -1 \]. Therefore, \[ (-i)^2 = 1 \times (-1) = -1 \]

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
To grasp complex numbers, you must understand the imaginary unit, denoted as \(i\). The imaginary unit has a defining property: \(i^2 = -1\). With this fundamental concept, you can handle various calculations involving complex numbers that go beyond real numbers. Here are some key points about the imaginary unit:
  • \(i^1 = i\)
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)
These values cycle every four powers, so knowing this cycle is important for simplifying powers of \(i\). Understanding \(i\) prepares you for more complex manipulations of complex numbers.
complex number multiplication
Multiplying complex numbers can be tricky but is easier with practice. When you multiply complex numbers, you often use the distributive property: ewline If you multiply \((a + bi)\) and \((c + di)\), the steps are as follows:
  • Distribute \(a\): \(a \times c + a \times di\)
  • Distribute \(bi\): \(bi \times c + bi \times di\)
This expands to: ewline \[(a + bi)(c + di) = ac + adi + bci + bidi\] Remember, \(i^2 = -1\), so it simplifies to: ewline \[ac + adi + bci + bdi^2 = ac + adi + bci - bd\] Finally, combine like terms: ewline \[ac - bd + (ad + bc)i\] For the specific example in the exercise, multiplying \((-i)^2\), you have terms like \(-1 \times i\), which simplifies accordingly.
properties of exponents
Knowing exponent rules is crucial when dealing with powers of complex numbers. Here are fundamental properties:
  • \((ab)^n = a^n b^n\)
  • \((a^m)^n = a^{mn}\)
  • \(a^{m + n} = a^m \times a^n\)
  • \(a^0 = 1\) for any \(a eq 0\)
For complex numbers, apply these rules similarly. In the provided exercise, we see: ewline \((-i)^2\) simplified to \((-1 \times i)^2\). Using property one, this becomes: ewline \[(-1)^2 \times i^2 = 1 \times -1 = -1\] Practice makes perfect, so try out more exercises to become confident.

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

Multiply or divide as indicated. $$ \sqrt{-10} \cdot \sqrt{2} $$

Concept Check Let \(a=1\) and let \(b=64\) (a) Evaluate \(\sqrt{a}+\sqrt{b} .\) Then find \(\sqrt{a+b} .\) Are they equal? (b) Evaluate \(\sqrt[3]{a}+\sqrt[3]{b}\). Then find \(\sqrt[3]{a+b} .\) Are they equal? (c) Complete the following: In general, \(\sqrt[n]{a}+\sqrt[n]{b} \neq\) ________ based on the observations in parts (a) and (b) of this exercise.

Multiply or divide as indicated. $$ \sqrt{-3} \cdot \sqrt{11} $$

Find each quotient. $$ \frac{-1+5 i}{3+2 i} $$

Find each quotient. $$ \frac{2}{1+i} $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

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.