Problem 51 - Intermediate Algebra

Question

Multiply. Write all answers in the form \(a+b i.\) $$ 7(5-4 i) $$

Step-by-Step Solution

Answer

The answer is \(35 - 28i\).

Step 1: Distribute the Real Number

Multiply the real number 7 by each term inside the parentheses. First, multiply 7 by 5, which equals 35.

Step 2: Distribute with the Imaginary Number

Continue to distribute 7 by multiplying it with the imaginary term (-4i). This yields -28i.

Step 3: Combine Results in Standard Form

Combine the results from step 1 and step 2 to write the expression in the form of \(a + bi\). So, you have 35 - 28i.

Key Concepts

Multiplication of Complex Numbers

Understanding the multiplication of complex numbers opens doors to a wide range of applications in mathematics and beyond. When multiplying a complex number by a real number, you follow similar principles as you do with regular multiplication but with an additional layer—the imaginary unit. In the given exercise, we have to multiply the complex expression, which is presented as a quantity in parentheses,
7 and a binomial,
(5 - 4i).

Here's a quick breakdown of how multiplication works with complex numbers:

• First, multiply the real components: The real number 7 is multiplied by the real part of the complex number, which is 5.
• Second, multiply the real number by the imaginary part: In this case, 7 is also multiplied by the imaginary part, -4i.
• Combine the results: The final expression is written in the form of a complex number, which is standard as a + bi.

Practicing this method helps solidify your understanding, offering a straightforward approach to handling such problems.

Distributive Property

The distributive property is a cornerstone of algebra that simplifies how we approach multiplication across terms. It essentially states that a single term, or number, can be multiplied across an entire set of terms inside parentheses. Applying this rule with complex numbers ensures that no parts are overlooked.

In our example, it's crucial to carefully distribute the real number 7 across each term:

• Start with the real component: Multiply 7 by 5 to get 35. This focuses on distributing with the tangible or real parts only.
• Next, handle the imaginary component: 7 is multiplied by -4i, which gives -28i. This reflects the distribution to both parts of the complex number.

Remember, distribution requires equal treatment of all terms, ensuring we end up with a complete expression that combines both real and imaginary components seamlessly.

Imaginary Unit

The imaginary unit, denoted by i, is pivotal in the world of complex numbers. It’s defined as the square root of -1, which is not a number you'd find on the regular number line. However, it's incredibly useful in expressing numbers that are solutions to equations that have no real number solutions.

Here's what you need to know about the imaginary unit and how it interacts in math:

• The imaginary unit 'i' when squared gives -1. This property is essential for simplifying expressions that include i.
• When a real number is multiplied by 'i', it forms the imaginary part of a complex number. For instance, in the exercise, we handled -4i by evaluating it with a standard real number, following arithmetic rules.

Understanding the imaginary unit helps you tackle complex number problems more effectively. It adds depth to our numerical system, allowing the expression and solution of equations that wouldn’t otherwise be solvable with only real numbers.