Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Write each number in simplest form, without a negative radicand. $$ -\sqrt{-400} $$

Step-by-Step Solution

Answer

The simplest form is \(-20i\).

Step 1: Identify the Negative Sign

The expression given is \(-\sqrt{-400}\). The negative sign outside the square root indicates that the result will be negative, once we resolve the square root. So, temporarily, we can consider \(\sqrt{-400}\) without worrying about the negative sign for now.

Step 2: Rewrite with Imaginary Unit

Since the radicand is negative, \(-400\), we can express \(\sqrt{-400}\) using the imaginary unit \(i\), where \(i = \sqrt{-1}\). Thus, \(\sqrt{-400} = \sqrt{400} \cdot i\).

Step 3: Simplify the Square Root of Positive Part

Calculate \(\sqrt{400}\). Since \(400\) is a perfect square, \(\sqrt{400} = 20\). Now, combine this with \(i\) from Step 2, leading to \(\sqrt{-400} = 20i\).

Step 4: Combine with the Outside Negative Sign

Recall the external negative sign. Combine it with the result from Step 3. Thus, \(-\sqrt{-400} = -20i\). This is the simplest form without a negative radicand.

Key Concepts

Imaginary Unit

When dealing with complex numbers and negative radicands, the imaginary unit, denoted as \(i\), plays a crucial role. The imaginary unit is defined as \(i = \sqrt{-1}\). This definition facilitates the manipulation and simplification of negative under-the-root expressions, converting them to a form that is manageable and meaningful in the domain of complex numbers.
For example, suppose you have a radicand like \(-400\). Without the imaginary unit, finding the square root of a negative number isn't feasible within the realm of real numbers. However, by employing \(i\), we can express \( \sqrt{-400} \) as \( \sqrt{400} \cdot i \).
This allows us to treat the negative sign separately and handle the mathematical operations correctly, translating a traditionally unsolvable problem in real numbers into one that elegantly utilizes the imaginary number system.

Negative Radicand

Radicands with negative values can be intimidating at first glance because, traditionally, the square root of a negative number is not defined within the set of real numbers. This is where complex numbers become useful. By introducing the imaginary unit \(i\), negative radicands can be rewritten in a form that is easily calculable.
For instance, if you encounter \(\sqrt{-400}\), the strategy involves separating the negative sign by utilizing \(i\). This gives us \(\sqrt{400} \cdot i\), where \(\sqrt{400}\) is perfectly calculable.
Negative radicands, therefore, often serve as a gateway into the world of complex numbers, helping expand numerical operations beyond the limitations of real numbers. This transformation is essential in many areas of mathematics and engineering, where complex numbers are frequently applicable.

Simplifying Radicals

Simplifying radicals is essential as it helps in presenting the expression in its simplest, most digestible form. The key steps in this process involve breaking down the radicand into its component factors, particularly looking for perfect squares.

• First, focus on isolating any negative sign by using \(i\), if applicable.
• Then, compute the square root of any positive component systematically.

For \(-\sqrt{-400}\), we first notice that 400 is a perfect square, making \(\sqrt{400} = 20\). We use \(i\) for the negative part, transforming \(\sqrt{-400}\) into \(20i\).
Finally, incorporate any external factors, such as a negative sign in this case, resulting in \(-20i\). This cumulative approach simplifies radicals involving negative numbers into elegant expressions using imaginary numbers, ensuring clarity and accuracy in solving complex equations.