Problem 59 - Intermediate Algebra

Question

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (6-2 i)(3+i) $$

Step-by-Step Solution

Answer

The result is \(20 + 0i\).

Step 1: Apply the Distributive Property

Multiply each term in the first complex number \((6 - 2i)\) by each term in the second complex number \((3 + i)\). Apply the distributive property to get:\[(6)(3) + (6)(i) + (-2i)(3) + (-2i)(i)\].

Step 2: Calculate Each Term

Calculate each term from the distribution:- \((6)(3) = 18\)- \((6)(i) = 6i\)- \((-2i)(3) = -6i\)- \((-2i)(i) = -2i^2\)

Step 3: Simplify Using Properties of \(i\)

Recall that \(i^2 = -1\). Simplify \(-2i^2\) as follows:- \(-2(-1) = 2\). Combine the terms to get:\[18 + 6i - 6i + 2\].

Step 4: Combine Like Terms

Combine the real terms and the imaginary terms:- Real terms: \(18 + 2 = 20\)- Imaginary terms: \(6i - 6i = 0\). The result is:\[20 + 0i\].

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions and solve equations. It states that a term can be distributed across terms inside a parenthesis. In this exercise, we are distributing each term from the first complex number to each in the second. This means multiplying each part of one expression by every part of another expression.
For the expression \((6 - 2i)(3 + i)\), we distribute as follows:

• Multiply 6 by 3
• Multiply 6 by \(i\)
• Multiply \(-2i\) by 3
• Multiply \(-2i\) by \(i\)

This steps involves taking each component, whether real or imaginary, and combining all the possible products that can be formed. It’s basically expanding the brackets by making sure every element inside the first bracket multiplies every element inside the second.

Complex Multiplication

Complex multiplication involves multiplying complex numbers just like binomials, but paying attention to the imaginary unit \(i\). When multiplying two complex numbers, use the distributive property and then simplify using properties of the imaginary unit.
Remember in multiplication:

• Real parts multiply together,
• Imaginary parts multiply together,
• Real parts and imaginary parts mix as products.

For example, in \((6-2i)(3+i)\):

• Multiply the real number 6 by 3 to get 18.
• Multiply the real part 6 with imaginary \(i\) to get \(6i\).
• Multiply the imaginary part \(-2i\) with 3 to get \(-6i\).
• Multiply the imaginary parts \(-2i\) and \(i\) to get \(-2i^2\).

Each of these multiplications handles either a real or imaginary component, allowing you to transform complex expressions into a solvable format.

Imaginary Unit Properties

In complex numbers, the imaginary unit \(i\) is essential. It represents the square root of \(-1\). Understanding its properties is key to simplifying complex expressions. The most crucial property of \(i\) is its powers:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

These repeat in cycles. Always simplify higher powers using these. In our exercise, \(-2i^2\) simplifies to 2 because \(i^2 = -1\).
Thus, during simplification, replace \(i^2\) with \(-1\) to make calculations straightforward. This change turns complex calculations into simpler real expressions, aiding in straightforward resolution.

Simplification of Complex Expressions

Simplifying complex expressions involves combining and reducing terms to their simplest form. After using the distributive property and performing multiplication, we end up with several terms, either real or imaginary.
Start by identifying and merging like terms. Real parts of a complex expression stand on their own, while imaginary parts combine separately. In this exercise:

• The real terms are \(18\) and 2, giving us \(18 + 2 = 20\).
• The imaginary terms are \(6i\) and \(-6i\), which cancel out, resulting in 0.

Adding these together, the expression is simplified to \(20 + 0i\). Through this step-by-step reduction, we obtain the simplest form, reflecting the comprehensible outcome of manipulating complex numbers.