Problem 21 - A Graphical Approach to College Algebra

Question

Write each number in simplest form, without a negative radicand. $$\sqrt{-100}$$

Step-by-Step Solution

Answer

\(10i\)

Step 1: Identify the Negative Radicand

The expression given is \(\sqrt{-100} \). Notice the radicand is negative, which means we are dealing with an imaginary number. The square root of negative numbers involves imaginary numbers.

Step 2: Separate the Negative and Positive Components

We can rewrite the expression as \(\sqrt{-1} imes \sqrt{100} \), where \(\sqrt{-1}\) is the imaginary unit \(i\).

Step 3: Evaluate the Square Root of Positive Component

Calculate the square root of 100. Since 100 is a perfect square, \(\sqrt{100} = 10\).

Step 4: Multiply the Imaginary Unit by the Square Root

Combine the results of step 2 and step 3: \(i imes 10\), which simplifies to \(10i\).

Step 5: Express in Simplest Form

The expression \(10i\) is the simplest form of the original expression \(\sqrt{-100} \), as it has no negative radicand and is expressed in terms of the imaginary unit \(i\).

Key Concepts

Complex Numbers

Complex numbers are a fascinating concept in mathematics that extend the idea of traditional numbers. They include both a real and an imaginary part. The imaginary unit, often denoted by \( i \), is defined as \( \sqrt{-1} \). Because of this, \( i^2 \) equals \( -1 \).

Complex numbers are generally written in the form \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part. This is essential because it allows us to handle equations that do not have real solutions, like \( x^2 + 1 = 0 \). Such equations can be solved using complex numbers.

When you encounter a square root of a negative number, like \( \sqrt{-100} \), it implies an imaginary number. The solution involves separating it into its real and imaginary components, using that core property of \( i \). By rewriting \( \sqrt{-100} \) as \( \sqrt{100} \times \sqrt{-1} \), and knowing \( \sqrt{-1} = i \), you resolve this to a complex number \( 10i \). Thus, any negative square roots are easily simplified using the properties of complex numbers.

Radicals

Radicals are a mathematical notation used to indicate roots of numbers or expressions. The most common radical is the square root, represented by the symbol \( \sqrt{} \). However, there are other roots such as cube roots (\( \sqrt[3]{} \)) and fourth roots, etc.

Understanding radicals often involves simplifying them, which can mean expressing them in terms of simpler radicals or completely simplifying the expression. When dealing with negative radicands, like in \( \sqrt{-100} \), it involves introducing imaginary numbers. This helps in rewriting and simplifying what would otherwise be an undefined expression in the realm of real numbers.

In practical terms, when facing radicals particularly involving imaginary numbers, remember:

• Identify the radicand, which is the number under the radical.
• For negative radicands, utilize the imaginary unit \( i \).
• Simplify by handling both the real root and the imaginary component separately.

Radicals may seem tricky at first, but they become simpler with practice and familiarity.

Square Roots

Square roots are one of the most fundamental concepts in mathematics. They are defined as values that, when multiplied by themselves, produce the original number. For example, the square root of 100 is 10, because \( 10 \times 10 = 100 \).

However, when dealing with negative numbers under the square root, the situation changes significantly. In the context of real numbers, square roots of negative numbers do not exist. This is because no real number squared will give a negative result. This is where imaginary numbers and the unit \( i \) come into play.

To find the square root of a negative number like \( \sqrt{-100} \), you:

• Recognize the negative sign and apply the imaginary unit \( i \), turning the expression into \( \sqrt{100} \times \sqrt{-1} \).
• Calculate the square root of the positive part, \( 10 \), and then multiply by \( i \) to get \( 10i \).

This procedure transforms a potentially confusing expression into something manageable and opens the door to a broader understanding of numbers beyond the real number line.