Problem 109 - Elementary and Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i .\) $$ (2+i \sqrt{2})(3-i \sqrt{2}) $$

Step-by-Step Solution

Answer

The result is \(8\).

Step 1: Identify the Binomial Expressions

We have two complex numbers in the form of binomial expressions: \((2 + i \sqrt{2})\) and \((3 - i \sqrt{2})\).

Step 2: Identify the algebraic structure

Determine the type of algebraic problem.

Step 3: Apply algebraic techniques

Use factoring, expanding, or systematic methods.

Step 4: Simplify and solve

Simplify expressions and solve for unknowns.

Step 5: State the result

Write the final answer.

Key Concepts

Understanding Binomial Expressions

A binomial is an expression consisting of two terms. In the context of complex numbers, a binomial expression generally takes the form \(a + b\), where the terms involve constants and the imaginary unit, often represented by any form such as \(i\sqrt{2}\) or \(3 - i\sqrt{2}\). For complex numbers, each term of a binomial can include:

• Real numbers like 2 or 3.
• Imaginary parts such as \(i\sqrt{2}\).

In the given exercise, the expressions \( (2 + i \sqrt{2}) \) and \( (3 - i \sqrt{2}) \) are binomials since they each contain two distinct parts. This split allows us to apply different arithmetic operations independently on the real and imaginary components. Understanding binomial expressions enables us to manipulate complex numbers strategically, easily facilitating operations like multiplication or addition.

The Role of the Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in the study of complex numbers. It is defined by the property \(i^2 = -1\). This definition is crucial because it allows us to work with square roots of negative numbers in a meaningful way.

• The symbol \(i\) is used throughout algebra to represent the principal square root of \(-1\).
• In the exercise, \(i\sqrt{2}\) represents an imaginary term, where \(\sqrt{2}\) is real and \(i\) signifies the imaginary part.

With the imaginary unit, complex numbers can be represented in the form \(a + bi\), where \(a\) is the real component and \(bi\) represents the imaginary component. The imaginary unit simplifies the multiplication of complex numbers by enabling the conversion of the imaginary part into a familiar numerical format, allowing calculations to be performed smoothly.

Performing Complex Multiplication

Complex multiplication involves multiplying binomials which often include real and imaginary terms. For the expression \((2 + i \sqrt{2})(3 - i \sqrt{2})\), we perform the operation as follows:First, we apply the distributive property or FOIL method (First, Outer, Inner, Last):

• First: Multiply the first terms: \(2 \cdot 3 = 6\).
• Outer: Multiply the outer terms: \(2 \cdot (-i \sqrt{2}) = -2i \sqrt{2}\).
• Inner: Multiply the inner terms: \(i \sqrt{2} \cdot 3 = 3i \sqrt{2}\).
• Last: Multiply the last terms: \(i \sqrt{2} \cdot (-i \sqrt{2}) = -(i^2 \cdot 2) = -(-1 \cdot 2) = 2\).

Next, we combine these products:

• The real parts: \(6 + 2 = 8\).
• The imaginary parts: \(-2i \sqrt{2} + 3i \sqrt{2} = i \sqrt{2}\).

Finally, the result of the multiplication can be neatly written in standard form: \(8 + i \sqrt{2}\). This method ensures all components are properly accounted for, giving a clear and correct result for complex multiplication.