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Chapter 3: Problem 3

Simplify by using the imaginary unit \(i\). $$ \sqrt{-100} $$

Short Answer

Expert verified
\(\sqrt{-100} = 10i\)

Step by step solution

01

Recognize the Imaginary Unit

The imaginary unit, denoted by \(i\), is defined as \(i = \sqrt{-1}\). This allows us to express square roots of negative numbers using \(i\).
02

Apply the Imaginary Unit

Express \(\sqrt{-100}\) in terms of \(i\). Start by rewriting \(\sqrt{-100}\) as \(\sqrt{100 \times -1}\).
03

Simplify the Expression

Using the property \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\), rewrite \(\sqrt{100 \times -1}\) as \(\sqrt{100} \times \sqrt{-1}\). This becomes \(10 \times i\) since \(\sqrt{100} = 10\) and \(\sqrt{-1} = i\).
04

Final Answer

Thus, \(\sqrt{-100} = 10i\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
Simplifying expressions in mathematics involves reducing them to their simplest or most basic form. This often makes the expressions easier to understand and work with in calculations.
When dealing with expressions that include square roots, especially negatives, additional techniques are needed.
  • First, identify any negative signs inside the square root.
  • Use the properties of exponents and roots, such as \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\).
  • Apply any special definitional tools, such as the imaginary unit \(i\), to handle negatives under the square root efficiently.
This process turns complicated expressions into more manageable and often more meaningful representations.
Complex Numbers
Complex numbers are numbers that have both a real part and an imaginary part. They are written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.
  • Real part: The component of the complex number that does not accompany the \(i\). It's similar to the numbers you're already familiar with.
  • Imaginary part: This is the portion that includes \(i\). It's crucial for expressing square roots of negative numbers, as they cannot be expressed using just real numbers.
Complex numbers allow for greater flexibility in mathematics, enabling solutions to equations that have no real number solutions. They are used not just in mathematics, but also in engineering and physics to model real-world phenomena.
Imaginary Unit
The imaginary unit, denoted \(i\), is a mathematical concept that solves the problem of taking the square root of a negative number. By definition, \(i\) is equal to \(\sqrt{-1}\).
  • Using \(i\) allows us to extend the real number system to the complex number system.
  • Every imaginary number can be expressed as a multiple of \(i\), such as \(5i\) or \(-3i\).
  • In calculations, \(i^2\) equals \(-1\), simplifying powers of \(i\).
Introducing \(i\) expands our understanding of numbers, showing us that there is more beyond just the number line. This concept is foundational for any work involving complex numbers, providing a way to work with previously "impossible" numbers.

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Most popular questions from this chapter

Use a table to solve. \(\frac{1}{8} x^{2}+x+2 \geq 0\)

Give an example of a quadratic function that has only real zeros and an example of one that has only imaginary zeros. How do their graphs compare? Explain how to determine from a graph whether a quadratic function has real zeros.

Complete the following. (a) Write the equation as \(a x^{2}+b x+c=0\) with \(a>0\) (b) Calculate the discriminant \(b^{2}-4 a c\) and determine the number of real solutions. (c) Solve the equation. $$ 4 x=6+x^{2} $$

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