/*! 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

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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, \(\text{it becomes} \ 10i \).

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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.

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