Problem 30 - College Algebra with Modeling and Visualization

Question

Write the expression in standard form. $$ (-5)(-7+3 i) $$

Step-by-Step Solution

Answer

The expression in standard form is \(35 - 15i\).

Step 1: Apply the Distributive Property

To distribute (-5) across the complex number (-7 + 3i), apply the distributive property: (-5) imes (-7) + (-5) imes (3i).

Step 2: Calculate Each Term

Calculate the product of each distributed term: 1. (-5) imes (-7) = 35 2. (-5) imes (3i) = -15i. This gives us the expression: 35 - 15i.

Step 3: Write in Standard Form

The standard form for a complex number is a + bi, where a and b are real numbers. In this case, 35 - 15i is already in standard form: 35 + (-15)i, where a = 35 and b = -15.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that makes simplifying expressions easier. When you apply the distributive property, you multiply a single term by each term inside a bracket. This property is a useful tool when dealing with both real numbers and complex numbers.

• A complex number is often written in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit.
• The distributive property allows you to expand expressions by distributing a multiplying factor across the terms within a parenthesis.

To illustrate, let's consider \((-5)(-7 + 3i)\). By using the distributive property, you multiply \(-5\) by each of \(-7\) and \(3i\), resulting in two new terms. This step is crucial to simplifying and eventually finding the solution for expressions that include complex numbers.

Standard Form

When working with complex numbers, it is important to express the final result in standard form. The standard form of a complex number is written as \(a + bi\), where \(a\) and \(b\) are real numbers.

• The real part, \(a\), represents the real number in the expression.
• The imaginary part, \(bi\), includes both the coefficient \(b\) and the imaginary unit \(i\).

This format is handy because it clearly shows the real and imaginary components of the number. Using our earlier example, after distributing and simplifying \((-5)(-7) + (-5)(3i)\), we obtain \(35 - 15i\). Observing this expression, we see that it perfectly adheres to the standard form of a complex number where \(a = 35\) and \(b = -15\). It’s clear and organized, making it easier to work with complex numbers in future calculations.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a special number that doesn't quite fit into the world of real numbers. It is defined as \(i = \sqrt{-1}\). This unique property of \(i\) allows for the representation and operation of complex numbers.

• When squared, \(i^2\) equals \(-1\). This can lead to interesting results in calculations and is a foundational factor in complex numbers.
• Complex numbers are built on the combination of real numbers and multiples of \(i\), making them versatile for solving equations involving square roots of negative numbers.

In the expression \(35 - 15i\), \(i\) signifies the imaginary part, with \(-15\) being the coefficient of the imaginary unit. Understanding \(i\) is key to grasping complex number operations, as it allows you to work with expressions that would otherwise lack real solutions.