Problem 14 - College Algebra with Modeling and Visualization

Question

Simplify by using the imaginary unit \(i\). $$ \frac{2 \pm \sqrt{-8}}{4} $$

Step-by-Step Solution

Answer

\(\frac{1}{2} \pm \frac{\sqrt{2}}{2}i\).

Step 1: Identify the Imaginary Unit

Recognize that the square root of a negative number uses the imaginary unit. In this case, \(\sqrt{-8} = \sqrt{8} \cdot \sqrt{-1} = 2\sqrt{2}i\).

Step 2: Substitute the Imaginary Part

Substitute \(\sqrt{-8}\) with its expression in terms of \(i\). Thus, the expression becomes \(\frac{2 \pm 2\sqrt{2}i}{4}\).

Step 3: Separate the Real and Imaginary Parts

Separate the fraction into its real and imaginary parts: \(\frac{2}{4} \pm \frac{2\sqrt{2}i}{4}\).

Step 4: Simplify Each Part

Simplify each part of the expression. The real part becomes \(\frac{2}{4} = \frac{1}{2}\), and the imaginary part becomes \(\pm \frac{2\sqrt{2}i}{4} = \pm \frac{\sqrt{2}}{2}i\).

Step 5: Combine the Simplified Components

Combine the simplified real and imaginary parts to rewrite the expression as \(\frac{1}{2} \pm \frac{\sqrt{2}}{2}i\).

Key Concepts

Imaginary Unit

The concept of the imaginary unit is central to understanding complex numbers. When dealing with real numbers, you might recall encountering the square root of negative numbers and finding it impossible with traditional mathematics. Enter the imaginary unit, denoted as \(i\), which is defined as \(i = \sqrt{-1}\). This allows mathematicians to extend the real number system to include complex numbers. In practical terms, when you encounter \(\sqrt{-8}\), it can be broken down into \(\sqrt{8} \cdot \sqrt{-1}\), resulting in \(2\sqrt{2}i\). Now, we can manage calculations that include the square root of negative numbers with ease, applying the properties of imaginary numbers.

Simplification

Simplification helps in reducing complex algebraic expressions into a more manageable form. In the exercise, you began with the expression \(\frac{2 \pm \sqrt{-8}}{4}\). By understanding the role of the imaginary unit, after expressing \(\sqrt{-8}\) as \(2\sqrt{2}i\), it becomes important to substitute and simplify further.

Here's how simplification proceeds:

• Substitute \(\sqrt{-8}\) with \(2\sqrt{2}i\) to update the expression to \(\frac{2 \pm 2\sqrt{2}i}{4}\).
• Separate the expression into real and imaginary parts: \(\frac{2}{4}\) and \(\pm \frac{2\sqrt{2}i}{4}\).
• Simplify both segments. For the real part: \(\frac{2}{4} = \frac{1}{2}\). For the imaginary part: \(\pm \frac{2\sqrt{2}i}{4} = \pm \frac{\sqrt{2}}{2}i\).

Combining these simplified forms gives you \(\frac{1}{2} \pm \frac{\sqrt{2}}{2}i\), a cleaner, simplified version of the original complex expression.

Algebraic Expressions

Algebraic expressions are fundamental in mathematics, composed of variables, constants, and operations like addition, subtraction, multiplication, and division. In the context of the given exercise, these expressions include both real and imaginary components.

Understanding and manipulating such expressions necessitate comfort with combining like terms and adhering to algebraic rules. For complex numbers, it means handling the real and imaginary parts separately to simplify them as much as possible. Each part can be treated like a normal algebraic expression but remembering to incorporate \(i\) correctly:

• Real terms like \(\frac{2}{4}\) are straightforward to simplify.
• Imaginary terms such as \(\pm \frac{2\sqrt{2}i}{4}\) involve dealing with \(i\) as you would any variable, while applying the rules of multiplication and division.

By separating algebraic expressions into manageable parts, you gain the ability to simplify complex numbers and solve problems efficiently. This exercise showcased how intricate expressions can be broken down methodically, reinforcing basic algebraic principles.