Problem 20 - 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 \(-12 + 8i\).

Step 1: Expand the expression

To perform the operation, we first apply the distributive property, which states that the expression \((a+b)(c) = ac + bc\). Here, we will multiply \((2+3i)\) and \(4i\). Thus, \((2+3i)(4i) = 2(4i) + 3i(4i)\).

Step 2: Perform the multiplications

Carry out the multiplications from Step 1:- For \(2(4i)\), multiply 2 and \(4i\) to get \(8i\).- For \(3i(4i)\), multiply the components: 3 times 4 is 12, and \(i \times i = i^2\). Therefore, \(3i(4i) = 12i^2\).

Step 3: Simplify using \(i^2 = -1\)

Recall that the square of the imaginary unit \(i\) is \(-1\), i.e., \(i^2 = -1\). Substitute \(12i^2\) with \(12(-1)\), giving us \(-12\).

Step 4: Combine real and imaginary parts

Combine the real part and the imaginary part from our results. We have\(-12 + 8i\).This is already in the simplest form of a complex number, with the real part \(-12\) and the imaginary part \(8i\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to simplify expressions by distributing a single term across terms inside parentheses. This property is especially useful when dealing with complex numbers, as it enables the multiplication of two binomials or the distribution of a product over a sum.

In our exercise, we have the expression

• \((2+3i)(4i)\)

Using the distributive property, which states

• \((a+b)(c) = ac + bc\)

we can rewrite the expression as two separate products:

• \(2(4i) + 3i(4i)\)

This simplification breaks down complex operations into manageable steps, making it easier to handle each component separately. Understanding this concept is crucial as it forms the basis for further operations in algebra and calculus.

Imaginary Unit

The imaginary unit, often represented as \(i\), is a special number in mathematics that allows us to express numbers that are not on the real number line. The defining property of the imaginary unit is that its square is equal to

• \(i^2 = -1\)

This unique property is what makes complex numbers possible. In the exercise, the imaginary unit \(i\) appears within the multiplication

• \(3i(4i) = 12i^2\)

Understanding \(i\) is essential when dealing with complex numbers. It extends the real number system to include numbers like \(2 + 3i\), adding a new dimension of numerical representation. Without \(i\), we could not express or work with solutions to certain algebraic equations, such as those involving square roots of negative numbers.

Simplifying Expressions

Simplifying expressions, particularly with complex numbers, means reducing them to their simplest form, combining like terms where possible, and eliminating unnecessary components such as redundant parentheses or coefficients. In the provided exercise, we simplify the expression:

• \(8i + 12i^2\)

Using the property that \(i^2 = -1\), we replace \(12i^2\) with

• \(12(-1) = -12\)

The result is expressed as a complex number in standard form:

• \(-12 + 8i\)

This demonstrates proper simplification as it breaks down the expression into a clear real part and an imaginary part, making it easily interpretable. Knowing how to simplify complex expressions is invaluable in both pure and applied mathematics, allowing cleaner and more understandable solutions.