Problem 32 - A Graphical Approach to College Algebra

Question

Write each number in simplest form, without a negative radicand. $$i \sqrt{-16}$$

Step-by-Step Solution

Answer

The simplest form is \(4i\).

Step 1: Identify the Complex Component

The expression given is \(i \sqrt{-16}\). The key part here is dealing with the negative radicand \(-16\). Recall that the square root of a negative number involves the imaginary unit \(i\). Recognize that \(\sqrt{-1} = i\). Therefore, we write \(\sqrt{-16}\) as \(\sqrt{-1 \times 16} = \sqrt{-1} \times \sqrt{16} = i \sqrt{16}\).

Step 2: Simplify the Non-Negative Square Root

We now simplify \(\sqrt{16}\). Since \(16\) is a perfect square, \(\sqrt{16} = 4\). Thus, our expression becomes \(i \times 4\) after replacing the square root.

Step 3: Rearrange the Expression

Finally, write the expression in simplest form by rearranging it: \(i \times 4\) is written as \(4i\). This gives us the final simplified version of the original expression.

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in the world of complex numbers. It's used to represent the square root of negative one. In many practical scenarios, negative numbers inside a square root are non-intuitive, which makes \(i\) incredibly useful.

• By definition, \(i = \sqrt{-1}\).
• It allows us to express square roots of negative numbers as multiples of \(i\).

When working with complex numbers, \(i\) plays the role of transitioning a negative radicand into a workable mathematical expression. For example, to handle expressions like \(\sqrt{-16}\), we incorporate \(i\) to rewrite it as \(i \sqrt{16}\). This step-by-step breakdown allows us to simplify and make sense of these numeric expressions. Ultimately, understanding \(i\) is essential for simplifying and manipulating complex numbers.

Square Roots

Square roots are a basic mathematical operation that can be expanded when involving negative numbers. A square root asks: "What number, when multiplied by itself, gives the original number?" However, for negative numbers, we cannot find a real number solution because a negative times a negative results in a positive.To solve square roots of negative numbers:

• Separate the negative part: \(\sqrt{-a} = \sqrt{-1 \times a}\).
• Use the imaginary unit: Translate \(\sqrt{-1}\) into \(i\) and solve \(\sqrt{a}\) normally.

Consider \(\sqrt{-16}\), which translates to \(\sqrt{-1 \times 16}\). By applying our steps, this becomes \(i \sqrt{16}\). Since \(16\) is a perfect square, solving \(\sqrt{16}\) gives us \(4\), leading to \(i \times 4 = 4i\). Understanding the manipulation of square roots involving negative radicands is vital for simplifying expressions that include complex numbers.

Simplification of Expressions

Simplification of expressions, especially those involving complex numbers, ensures the expression is in its most concise form. An expression like \(i \sqrt{-16}\) can appear complex initially, but with basic simplification techniques becomes straightforward.Here is how you can simplify such expressions:

• Resolve negative square roots using \(i\): Convert \(\sqrt{-1\times 16}\) to \(i\sqrt{16}\).
• Simplify any perfect square: Solve \(\sqrt{16} = 4\).
• Multiply and rearrange: Combine components efficiently, transforming \(i \times 4\) into \(4i\).

The goal is to have an expression that's easy to read and interpret. This practice of simplification not only helps in understanding complex numeric forms but also ensures calculations are easier to manage. Using these steps consistently aids in mastering algebraic operations involving imaginary numbers.