Problem 27 - A Graphical Approach to College Algebra

Question

Write each number in simplest form, without a negative radicand. $$5+\sqrt{-4}$$

Step-by-Step Solution

Answer

The expression simplifies to \(5 + 2i\).

Step 1: Identify the Complex Part

The given expression is \(5 + \sqrt{-4}\). Here, \(\sqrt{-4}\) is the complex part because it has a negative radicand. To simplify, we need to express the square root of the negative number using the imaginary unit \(i\).

Step 2: Simplify the Radical

By definition, \(i\) is the square root of \(-1\), i.e., \(i = \sqrt{-1}\). Thus, \(\sqrt{-4} = \sqrt{4} \times \sqrt{-1} = 2i\).

Step 3: Substitute and Simplify

Substitute the simplified form of \(\sqrt{-4}\) back into the original expression: \(5 + 2i\). This gives us the final simplified form of the expression.

Key Concepts

Understanding the Imaginary Unit

When dealing with complex numbers, the imaginary unit, denoted by the symbol \(i\), comes into play. The imaginary unit is defined as the square root of negative one, \(i = \sqrt{-1}\). This is essential because in the real number system, we cannot have a square root of a negative number. But in complex numbers, the imaginary unit is key to expressing such values.

• It helps simplify expressions with negative radicands.
• The imaginary unit \(i\) is crucial for forming complex numbers like \(5 + 2i\), where \(5\) is the real part and \(2i\) is the imaginary part.

Understanding \(i\) allows us to solve equations and express numbers that involve square roots of negative numbers.

Technique of Simplifying Radicals

Simplifying radicals involves breaking down the expression to its simplest form. When the radicand (the number under the radical symbol) is negative, as seen in \(\sqrt{-4}\), we first recognize that we need to employ the imaginary unit \(i\).
The key idea here is to separate the radical into a product of the square root of its positive counterpart with the imaginary unit. For example, \(\sqrt{-4} = \sqrt{4} \times \sqrt{-1}\). Simplifying this gives us \(2i\), where \(2\) is the square root of \(4\) and \(\sqrt{-1} = i\).
This method allows us to convert expressions with negative radicands into a format compatible with complex numbers, making them easier to work with and understand.

Dealing with a Negative Radicand

A negative radicand indicates the involvement of the imaginary unit. For example, in \(\sqrt{-4}\), the negative sign tells us that this isn't a standard radical. Instead, it's a complex number waiting to be simplified.

• Recognize that you need to involve \(i\), the imaginary unit.
• Convert the expression by separating the negative part as \(\sqrt{-4} = \sqrt{4} \times \sqrt{-1}\).
• This transforms the expression to \(2i\), making it part of the complex number \(5 + 2i\).

Working with negative radicands allows us to simplify expressions using the rules of complex numbers and enhances our understanding of how negative values can be represented in mathematics.