Problem 24 - College Algebra with Corequisite Support

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 \(11 + 10i\).

Step 1: Distribute the Terms

To multiply \((2+3i)(4-i)\), apply the distributive property: \[(2+3i)(4-i) = 2 \cdot 4 + 2 \cdot (-i) + 3i \cdot 4 + 3i \cdot (-i)\]

Step 2: Perform the Multiplications

Calculate each term from the distribution performed in Step 1:- \(2 \cdot 4 = 8 \)- \(2 \cdot (-i) = -2i\)- \(3i \cdot 4 = 12i\)- \(3i \cdot (-i) = -3i^2\)

Step 3: Simplify the Imaginary Unit

Recall that \(i^2 = -1\). Therefore, \(-3i^2 = -3(-1) = 3\)

Step 4: Combine Like Terms

Combine the real and imaginary parts:- Real part: \(8 + 3 = 11\)- Imaginary part: \(-2i + 12i = 10i\) The result is \(11 + 10i\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in mathematics, often used to simplify expressions and perform multiplications within parentheses. When we encounter an expression like

• a sum or difference multiplied by another expression, the distributive property allows us to "distribute" this outer term across each term within the parentheses.

By applying this property to complex numbers, the operation becomes more accessible. Consider multiplying two binomials such as

• given in the exercise \((2+3i)(4-i)\).

To simplify this expression, each term in the first complex number is multiplied by every term in the second complex number:

• First, multiply \(2\) by \(4\), then by \(-i\), yielding the terms \(8\) and \(-2i\).
• Next, multiply \(3i\) by \(4\), resulting in \(12i\), and then by \(-i\), resulting in \(-3i^2\).

Either way, you break it down, the distributive property helps to spread terms evenly, ensuring all parts are multiplied correctly.

Imaginary Unit

The imaginary unit, denoted as \(i\), is defined as the square root of \(-1\). This means that \(i^2 = -1\), a crucial identity to remember when working with complex numbers. The imaginary unit allows us to expand our number system beyond real numbers.
For example, the imaginary unit allows us to solve equations that don't have solutions among the real numbers. In the exercise, the term \(-3i^2\) appears, which comes into play when simplifying expressions involving \(i\).

• When we substitute \(i^2 = -1\) into \(-3i^2\), this becomes \(-3(-1)\),
• which simplifies to \(3\).

Dealing with \(i\) might initially seem tricky, but remembering the simple equation \(i^2 = -1\) assists significantly. It helps transform otherwise hard-to-manage expressions into simpler real or imaginary results.

Simplifying Expressions

When simplifying expressions, particularly those involving complex numbers, combining like terms is a critical step. Once you've completed distributing and multiplying, like we've done with each term in the complex binomial

• \((2+3i)(4-i)\),

you'll often end up with both real numbers and imaginary numbers.

• Start by grouping all real numbers together.
• In the exercise solution, we combined \(8\) and the \(3\) obtained from \(-3i^2\), adding them up to get the real part.
• Similarly, group the imaginary terms. The combined terms of \(-2i\) and \(12i\) in our exercise yield \(10i\).

After combining your terms accordingly, the expression is simplified into a standard form of a complex number, \(a + bi\), such as the final answer \(11 + 10i\) in our case. Maintaining clarity in each step ensures an accurate and simplified result.