Problem 36 - Intermediate Algebra

Question

Multiply. Write the product in the form \(a+b i .\) See Example 4. $$ 5 i(4-7 i) $$

Step-by-Step Solution

Answer

The product is \(35 + 20i\).

Step 1: Understand the Expression

The expression is given as \(5i(4 - 7i)\). We need to distribute \(5i\) across the terms in the parentheses to find the product in the form \(a + bi\).

Step 2: Distribute \(5i\) to Each Term

Distribute \(5i\) to each term inside the parentheses: \(5i \times 4\) and \(5i \times (-7i)\).

Step 3: Multiply \(5i\) by 4

Calculate \(5i \times 4 = 20i\). This is the real coefficient part of the imaginary component.

Step 4: Multiply \(5i\) by \(-7i\)

Calculate \(5i \times (-7i) = -35i^2\). Since \(i^2 = -1\), this simplifies to \(-35(-1) = 35\).

Step 5: Combine Like Terms

After distributing and simplifying, the terms are \(35 + 20i\). so the product is \(35 + 20i\).

Key Concepts

Imaginary Numbers

An imaginary number is a type of complex number wherein the real part is zero. To better understand, consider the imaginary unit, denoted as \(i\). The imaginary unit has a unique property: when it is squared, it becomes \(i^2 = -1\).
Imaginary numbers such as \(5i\) are obtained by multiplying a real number with the unit \(i\). For instance, in the expression \(5i(4 - 7i)\), \(5i\) represents an imaginary number. It acts like a coefficient in the multiplication process. This particular notation is pivotal in complex number computations as it aids in facilitating operations on complex numbers. By understanding imaginary numbers, you can easily tackle exercises that involve complex numbers.

Multiplication of Complex Numbers

When it comes to multiplying complex numbers, an organized approach is crucial. The process frequently involves the application of the distributive property, as seen in the task of multiplying \(5i(4 - 7i)\). Here's a breakdown:

• First, distribute the number outside the parentheses—in this case, \(5i\)—to each term inside the parentheses.
• This action leads to two separate multiplications: \(5i imes 4\) and \(5i imes (-7i)\).
• The multiplication \(5i imes 4\) results in \(20i\).
• The multiplication \(5i imes (-7i)\) results in \(-35i^2\), which is simplified to \(35\) since \(i^2 = -1\).

By utilizing this methodical approach, you derive an answer in the form \(a + bi\), where both real and imaginary components are neatly organized.

Distributive Property

The distributive property is a key concept when working with algebraic expressions involving both real and complex numbers. This property states that you must "distribute" or "multiply out" the term outside the parenthesis to every term inside. For example, in the expression \(5i(4-7i)\):

• The \(5i\) is distributed to the \(4\) and the \(-7i\) separately.
• This results in the two products: \(5i \times 4 = 20i\) and \(5i \times (-7i) = -35i^2\).
• The application of the distributive property ensures that each term within the parentheses is accurately multiplied by \(5i\).

Employing this property efficiently emphasizes the importance of methodically breaking down multiplications into simpler parts. It allows ensuring each component is appropriately addressed for accurate outcomes in complex number calculations.