Problem 22 - College Algebra

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 simplified complex number is \(30 - 10i\).

Step 1: Understand the Problem

We are given a complex number expression \((6-2i)(5)\). This means we need to multiply the complex number \((6-2i)\) by \(5\).

Step 2: Apply the Multiplicative Property

To multiply a complex number by a real number, distribute the real number to both the real and imaginary parts of the complex number. Calculate: \((6-2i) \times 5 = 6 \times 5 + (-2i) \times 5\).

Step 3: Calculate Each Part

Perform the multiplication for each part:1. \(6 \times 5 = 30\)2. \(-2i \times 5 = -10i\)

Step 4: Combine the Results

Combine the results from the real and imaginary parts: The complex number is \(30 - 10i\).

Key Concepts

Imaginary Numbers

Imaginary numbers are a fundamental part of complex numbers. They are denoted by the symbol \(i\), which represents the square root of \(-1\). This means \(i^2 = -1\).
Imaginary numbers allow us to extend the real number system and deal with quantities that involve the square root of negative numbers, which are not possible to express with real numbers alone.
In our exercise, the imaginary part is represented by \(-2i\) in the expression \((6-2i)\). Here, \(-2\) is the coefficient of \(i\), indicating it is scaled by this factor.

• This coefficient multiplies with \(i\) to form the imaginary component of a complex number.
• Real numbers, when combined with these imaginary units, create complex numbers.

Multiplication of Complex Numbers

Multiplying complex numbers involves combining both the real and imaginary components. This is done using the distributive property, similar to multiplying in algebra. In our problem, we were multiplying the complex number \((6-2i)\) by the real number \(5\).
First, distribute the real number \(5\) to each component of \(6-2i\):

• Multiply the real part: \(6 \times 5 = 30\).
• Multiply the imaginary part: \(-2i \times 5 = -10i\).

The result is

• a new complex number consisting of these two parts combined.

This cleanly aligns with how polynomials are multiplied, reinforcing the idea that complex numbers can be manipulated similarly to algebraic expressions.

Simplification of Complex Expressions

Simplification of complex expressions means combining like terms to express the number in its most basic form. The main goal is to present a complex number in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.
In our case, after distributing \(5\) to \((6-2i)\), we get two results: \(30\) and \(-10i\). To simplify, you simply combine these into one expression:

• You write it as: \(30 - 10i\) where \(30\) is the real part and \(-10i\) is the imaginary part.

This yields the simplified version of our original expression, providing a clear and concise result. Simplification means making it easy to understand and manipulate in further calculations.