/*! 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 \(\sqrt{-100}\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 6

\(\sqrt{-100}\)

Short Answer

Expert verified
10i

Step by step solution

01

Recognize the Problem

Identify that \(\text{you need to find the square root of a negative number,} \ \sqrt{-100}\).
02

Understand the Concept of Imaginary Numbers

Recall that the square root of a negative number involves imaginary numbers where \(i\) is defined as \(i = \sqrt{-1}\). Therefore, \( \sqrt{-a} = i \sqrt{a} \).
03

Apply the Concept

Rewrite \(\text{ the expression using imaginary numbers:} \ \sqrt{-100} = \sqrt{-1\cdot100} = \sqrt{-1} \cdot \sqrt{100} \).
04

Simplify

Simplify using the definitions \(\text{of imaginary numbers:} \ \sqrt{-1} = i \text{ and} \ \sqrt{100} = 10\). Thus, \(\text{it becomes} \ 10i \).

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.

Understanding Square Roots
When dealing with square roots, you are essentially looking for a number that, when multiplied by itself, equals the original number. If you want to find the square root of 25, you need to find a number that when multiplied by itself equals 25. Here, the answer is 5, because \(5 \times 5 = 25\). However, square roots of negative numbers are not that straightforward. Since no real number, when squared, gives a negative result, we use the concept of imaginary numbers to solve these problems.
Dealing with Negative Numbers in Square Roots
When we come across the square root of a negative number, such as \-100, we need to take a different approach. Normally, taking the square root of a negative number is not possible in the set of real numbers. This is where imaginary numbers come in handy. An imaginary number is expressed with the symbol \(i\), where \(i = \sqrt{-1}\). With this, any negative number can now be dealt with by separating it into its positive counterpart and \(i\). For instance, \-100 can be written as \(\sqrt{-100} = \sqrt{-1 \cdot 100} = \sqrt{-1} \cdot \sqrt{100}\).
Simplifying Complex Numbers
After rewriting the negative square root using imaginary numbers, the next step is simplification. Take \-100 as an example, first express it as \(\sqrt{-100} = \sqrt{-1 \cdot 100}\). Then, split it into its real and imaginary components: \(\sqrt{-1} \cdot \sqrt{100}\). Knowing that \(\sqrt{-1} = i\) and \(\sqrt{100} = 10\), you can replace these terms to get \(10\i\). This represents the solution in the realm of complex numbers, where every complex number can be written in the form \(a + bi\) with \(a\) and \(b\) being real numbers and \(i\) the imaginary unit. In our case, \sqrt{-100} simplifies to \(10i\), which is a basic form of a complex number.

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

\(\sqrt{3^{2}-4(2)(-8)}\)

\(\sqrt{3^{2}-4(8)(-1)}\)

\(m^{2}+24 m+167=0\)

Explain why the square root of a negative number cannot be a real number.

If the lead coefficient of \(y=a x^{2}+b x+c\) is a positive number, does the graph of the equation open up or open down?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

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.