/*! 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 2 Simplify each expression to a si... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 2

Simplify each expression to a single complex number. $$ \sqrt{-16} $$

Short Answer

Expert verified
The simplified expression is \( 4i \).

Step by step solution

01

Identify the Expression as a Complex Number

The expression \( \sqrt{-16} \) can be rewritten by recognizing that the square root of a negative number involves complex numbers. We know that \( i = \sqrt{-1} \), where \( i \) is the imaginary unit.
02

Rewrite the Expression Using the Imaginary Unit

Rewrite \( \sqrt{-16} \) as \( \sqrt{16} \times \sqrt{-1} \). This uses the fact that \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \).
03

Simplify the Real Part

Simplify \( \sqrt{16} \) to get 4. So, \( \sqrt{16} \times \sqrt{-1} = 4 \times \sqrt{-1} \).
04

Replace \( \sqrt{-1} \) with \( i \)

Substitute \( \sqrt{-1} \) with \( i \). Therefore, \( 4 \times \sqrt{-1} = 4i \).

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 imaginary unit, denoted as \( i \), is essential in dealing with complex numbers. It is defined by the equation \( i^2 = -1 \). This definition allows us to handle square roots of negative numbers, which don't have real solutions.
  • The main property of \( i \) is that it represents \( \sqrt{-1} \).
  • By using \( i \), we can express complex numbers, which are in the form \( a + bi \), where \( a \) and \( b \) are real numbers.
Understanding \( i \) is crucial when working with complex numbers, as it helps transform expressions involving the square root of negative numbers into manageable forms.
Simplification
Simplification is about transforming expressions into their simplest form. When working with complex numbers, this often involves reducing expressions to make them easier to understand and work with.
In our original exercise, the expression \( \sqrt{-16} \) was simplified by recognizing that it could be rewritten in terms of the imaginary unit \( i \).
  • The expression \( \sqrt{-16} \) was broken down into \( \sqrt{16} \times \sqrt{-1} \).
  • By simplifying \( \sqrt{16} \) to 4, the expression reduces to \( 4 \times \sqrt{-1} \).
  • Finally, substituting \( \sqrt{-1} \) with \( i \) gives \( 4i \).
This process highlights the importance of each step, ensuring clarity and comprehension in dealing with complex expressions.
Square Root of Negative Numbers
Understanding the square root of negative numbers might seem tricky at first, but it becomes straightforward when using complex numbers. Normally, the square root function is only defined for positive numbers. However, with the help of the imaginary unit, we can extend this to all real numbers.
For a negative number \( -a \), the square root can be expressed as:
  • \( \sqrt{-a} = i \sqrt{a} \)
The original problem involved \( \sqrt{-16} \). By recognizing that this is equal to \( \sqrt{16} \times \sqrt{-1} \), we used the imaginary unit to simplify
to \( 4i \).
By understanding how the square root of negative numbers relates to \( i \), we can more effectively handle and simplify these expressions in mathematics.

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

Find the dot product of each pair of vectors. $$ \langle 0,4\rangle ;\langle-3,0\rangle $$

Rewrite each complex number into polar \(r e^{1 \theta}\) form. $$ -2+3 i $$

Rewrite each complex number into polar \(r e^{1 \theta}\) form. $$ 4+7 i $$

Find the magnitude of the projection of \(\langle 2,7\rangle \) onto \(\langle 4,5\rangle .\)

Compute each of the following, simplifying the result into \(a+b i\) form. $$ \sqrt{-3+3 i} $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Decision 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.