Problem 79 - Elementary and Intermediate Algebra

Question

Simplify. Write the result in the form \(a+b i .\) \((3+4 i)(2-3 i)\)

Step-by-Step Solution

Answer

The simplified result is \(18 - i\).

Step 1: Distribute the Terms

We start by using the distributive property, also known as the FOIL method for multiplying complex numbers:\( (3+4i)(2-3i) = 3(2) + 3(-3i) + 4i(2) + 4i(-3i) \)

Step 2: Calculate Each Term

Calculate each of the four products:- The first term: \(3 \times 2 = 6\)- The outer term: \(3 \times -3i = -9i\)- The inner term: \(4i \times 2 = 8i\)- The last term: \(4i \times -3i = -12i^2\)

Step 3: Combine Like Terms

Before combining, remember that \(i^2 = -1\). Thus, \(-12i^2 = 12\). Now, combine the real numbers and the imaginary numbers:- Real numbers: \(6 + 12 = 18\)- Imaginary numbers: \(-9i + 8i = -i\)

Step 4: Write the Result in Standard Form

Combine the results from Step 3 to write in the form \(a + bi\):\( 18 - i \)

Key Concepts

Complex Number Multiplication

Multiplying complex numbers might seem tricky, but it's essentially an extension of the distributive property used in simple algebra. Each complex number has a real number part and an imaginary number part. For instance, in

• The number 3 is the 'real part' of 3 + 4i.
• 4i is the 'imaginary part'.
• Similarly, in 2 - 3i, 2 is real, and -3i is imaginary.

When multiplying two complex numbers, we treat them like binomials and use techniques similar to multiplying two algebraic expressions. This involves multiplying each part of one complex number by each part of the other, which leads us to explore more about the 'FOIL method' next.

FOIL Method

The FOIL Method stands for First, Outer, Inner, and Last. It's a technique used for multiplying two binomials. In the context of complex numbers, the expression offers a perfect scenario to apply the FOIL method:

• **First**: Multiply the first terms of each binomial: \(3 \times 2 = 6\).
• **Outer**: Multiply the outer terms of the expression: \(3 \times -3i = -9i\).
• **Inner**: Multiply the inner terms: \(4i \times 2 = 8i\).
• **Last**: Multiply the last terms: \(4i \times -3i = -12i^2\).

The FOIL method helps break down the multiplication into manageable parts, each representing a unique aspect of the complex numbers.

Distributive Property

The distributive property, a cornerstone in mathematics, is key to multiplying complex numbers. It says that a term outside a parenthesis can be distributed across each term inside the parenthesis. This is a universal principle applied here with complex numbers and simplifies the multiplication process.For example, with

• The entire expression can be expanded: \((3 + 4i)\) is distributed to each term in \((2 - 3i)\).

This allows us to comprehensively look at all possible combinations of terms, ensuring no component is overlooked. The process is similar to distributing a factor across a sum or difference, making it fundamental to operations with complex numbers.

Imaginary Unit

An essential part of understanding complex numbers is grasping the concept of the imaginary unit, denoted \(i\). The defining property of \(i\) is that \(i^2 = -1\). This is crucial because it distinguishes imaginary numbers from other numerical sets.In the context of our example, the term \(4i \times -3i = -12i^2\) simplifies further. Since \(i^2 = -1\), we substitute to get:

• \(-12i^2 = -12(-1) = 12\).

Recognizing the properties of \(i\) turns potentially complicated operations into straightforward calculations, converting imaginary components into a real one.