Problem 20 - College Algebra

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ (2+3 i)(4 i) $$

Step-by-Step Solution

Answer

The simplified complex number is \(-12 + 8i\).

Step 1: Multiply Each Term

To multiply complex numbers, we apply the distributive property. Start by multiplying the real and imaginary components of the binomial \((2 + 3i)\) by the complex number \(4i\). This gives us:\[(2 \times 4i) + (3i \times 4i)\]

Step 2: Simplify the Real Parts

Calculate the multiplication of real components with imaginary parts: - \(2 \times 4i = 8i\)

Step 3: Simplify the Imaginary Parts

Now, calculate the multiplication of imaginary components:- \(3i \times 4i = 12i^2\)Recall that \(i^2 = -1\), so:- \(12i^2 = 12(-1) = -12\)

Step 4: Combine Real and Imaginary Parts

Combine the results from the previous steps:- Real part: \(-12\)- Imaginary part: \(8i\)Thus, the simplified expression is:\(-12 + 8i\)

Key Concepts

Imaginary Numbers

Imaginary numbers are an exciting concept in mathematics. They extend the idea of numbers beyond the real number line. While real numbers can denote quantities, imaginary numbers are useful in various fields, including engineering and physics.

Imaginary numbers are based on the imaginary unit, denoted by \(i\), which is defined as the square root of \(-1\). This means \(i^2 = -1\). Imaginary numbers help in solving equations that do not have real solutions. For example, solving the equation \(x^2 + 1 = 0\) leads to the solution \(x = i\).

When we combine real numbers with imaginary numbers, we get complex numbers, typically written in the form \(a + bi\), where \(a\) and \(b\) are real numbers.

Distributive Property

The distributive property is a helpful tool when multiplying complex numbers. It allows us to ensure each part of one number is multiplied by each part of the other number. This property states that for any numbers \(a\), \(b\), and \(c\), the equation \(a(b + c) = ab + ac\) holds.

In the exercise given, we used the distributive property to multiply \((2 + 3i)\) by \(4i\). By distributing, we calculated:

• \(2 \times 4i\) gives us an imaginary part, \(8i\).
• \(3i \times 4i\) gives us \(12i^2\), which involves further simplification.

This approach neatly organizes the multiplication process and supports accurate simplification of complex numbers.

i Squared Equals Negative One

A key property of imaginary numbers is that \(i^2 = -1\). This is crucial when simplifying expressions involving imaginary numbers. It stems from the definition of the imaginary unit \(i\) as \(\sqrt{-1}\).

In the exercise, we encountered \(3i \times 4i\), resulting in \(12i^2\). Recognizing that \(i^2 = -1\) allows us to transform \(12i^2\) into \(12 \times (-1)\), simplifying it to \(-12\).

This step is vital because it lets us switch from an expression with \(i^2\) to a real number, allowing us to combine real and imaginary components into a simplified complex number. Understanding \(i^2 = -1\) is essential for simplifying and solving equations involving complex numbers.