Problem 3 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(i\).

Step 2: Apply the Imaginary Unit

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

Step 3: 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 \(\sqrt{-1} = i\).

Step 4: Final Answer

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

Key Concepts

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.