/*! 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 9 Express in terms of \(i\). $$ ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 9

Express in terms of \(i\). $$ \sqrt{-100} $$

Short Answer

Expert verified
The expression \( \sqrt{-100} \) can be expressed as \( 10i \).

Step by step solution

01

Understand the square root of a negative number

The square root of a negative number involves an imaginary unit. The imaginary unit is denoted by the symbol \( i \), where \( i = \sqrt{-1} \).
02

Rewrite the expression inside the square root

The given expression is \( \sqrt{-100} \). Rewrite \( -100 \) as \( (-1) \times 100 \). This gives us \( \sqrt{(-1) \times 100} \).
03

Use the property of square roots

Apply the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). Thus, \( \sqrt{(-1) \times 100} = \sqrt{-1} \times \sqrt{100} \).
04

Simplify the square root parts separately

We know \( \sqrt{-1} = i \). Also, \( \sqrt{100} = 10 \). Therefore, \( \sqrt{-100} = i \times 10 = 10i \).

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
Square roots help us find a number which, when multiplied by itself, yields the original number. For example, the square root of 9 is 3 because 3 times 3 equals 9.
To express this mathematically, we write \( \sqrt{9} = 3 \).
Let's revisit the concept using the formula \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). This property allows us to break down more complex square roots, like in the problem where \( \sqrt{-100} = \sqrt{(-1) \times 100} \), into simpler parts.
Here, \( \sqrt{(-1) \times 100} = \sqrt{(-1)} \times \sqrt{100} \). Breaking it into these parts simplifies our work.
Handling Negative Numbers
When dealing with negative numbers inside a square root, we need to understand the nature of these numbers.
A negative number means that the value is less than zero. For example, -3 is a negative number because it is three units less than zero.
In standard mathematics, we can't take the square root of a negative number because there's no real number which, when multiplied by itself, results in a negative value.
To comprehensively deal with negative square roots, mathematicians introduced the concept of imaginary numbers.
Introducing the Imaginary Unit
The imaginary unit, denoted by the symbol \(i\), is a special mathematical concept used for the square roots of negative numbers. The fundamental definition is \(i = \sqrt{-1}\).
This allows us to simplify square roots of negative numbers. For instance, the square root of -1 becomes i, and hence the square root of -100 becomes \( \sqrt{-100} = \sqrt{(-1) \times 100} = \sqrt{-1} \times \sqrt{100} = i \times 10 = 10i \).
Using i lets us handle complex problems and makes it easier to understand and work with numbers that step outside of the real number system.

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

Nadif incorrectly writes $$ \sqrt[5]{x^{2}} \cdot \sqrt{x^{3}}=x^{2 / 5} \cdot x^{3 / 2}=\sqrt[5]{x^{3}} $$ What mistake do you suspect he is making?

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. Write the answer using radical notation. $$ \frac{\sqrt[4]{x^{2} y^{3}}}{\sqrt[3]{x y}} $$

Consider the function g given by $$g(z)=\frac{z^{4}-z^{2}}{z-1}$$ Find \(g(3 i)\)

Simplify. $$\frac{i^{5}+i^{6}+i^{7}+i^{8}}{(1-i)^{4}}$$

Simplify. $$(-i)^{6}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Statistics

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.