Problem 22 - College Algebra with Corequisite Support

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ (6-2 i)(5) $$

Step-by-Step Solution

Answer

The result is \(30 - 10i\).

Step 1: Distribute the Real Number

Multiply the real number, 5, by both the real and imaginary parts of the complex number. This is done by distributing 5 to both terms in the complex number (6 and -2i). The operations are as follows: \[5 \times 6 = 30\] \[5 \times (-2i) = -10i\]

Step 2: Combine the Results

Write the products obtained in Step 1 together to form a new expression. The expression becomes:\[30 - 10i\]

Step 3: Simplify the Complex Number

Ensure the expression is written in a simplified complex form, that is, in the form \(a + bi\), where \(a\) and \(b\) are real numbers. In this exercise, it is already simplified:\[30 - 10i\]

Key Concepts

Multiplication of Complex Numbers

Multiplying complex numbers involves some rules that are a little different from multiplying regular numbers. It’s important to understand how to handle both the real and imaginary parts. When you multiply a complex number by another value, you multiply each part separately. For instance, in our example, with the expression

• (6 - 2i)(5)

we multiply the complex number

• 6 (the real part)
• -2i (the imaginary part)

by 5. This operation is straightforward as it follows standard multiplication, just applied to both parts of the complex number. Following the steps carefully ensures you don't mix the real and imaginary parts, which would make the calculation incorrect.

Distributive Property

The distributive property helps us handle expressions with multiple terms being multiplied. In this context, it involves distributing the multiplication over addition or subtraction inside parentheses. With complex numbers, we use it to distribute a multiplier over each part individually. Let's see how:

• The real part: Multiply 5 by 6 giving us 30.
• The imaginary part: Multiply 5 by -2i resulting in -10i.

This property divides the multiplication operation into segments that are easier to manage. Applying the distributive property makes solving multiplication problems involving complex numbers a much easier task. This systematic approach preserves the order of operations, making complex multiplication orderly and less error-prone.

Imaginary Unit

The imaginary unit, often denoted by the symbol \(i\), is a fundamental concept in complex numbers. Its defining characteristic is that \(i^2 = -1\). This property makes imaginary numbers unique and different from real numbers. When calculating with imaginary numbers within a complex expression, it’s essential to keep this special rule in mind. In the exercise we tackled, the imaginary part was

• -2i.

Even though we multiplied by 5, the imaginary unit \(i\) kept its properties. This unit helps us depict numbers that aren’t part of the regular real number system but still have practical applications, like in engineering and physics. Understanding this concept helps make solving complex number problems easier by recognizing that handling \(i\) follows its own set of rules.