Problem 39 - Algebra A Combined Function

Question

Multiply. Write your answers in the form \(a+b i\). $$ -3 i(-1+9 i) $$

Step-by-Step Solution

Answer

\(27 + 3i\)

Step 1: Apply the Distributive Property

To solve this exercise, we start by applying the distributive property to multiply \(-3i(-1+9i)\).This means we need to distribute \(-3i\) to each term inside the parentheses:\(-3i imes (-1) + (-3i) imes (9i)\).

Step 2: Multiply Each Term

Now multiply \(-3i\) by each term:1. For \(-3i imes (-1)\): we get \(3i\).2. For \((-3i) imes (9i)\): remember that \(i^2 = -1\). Thus, \((-3) imes 9 imes i^2 = -27 imes (-1) = 27\).

Step 3: Combine the Results

Combine the results from the multiplication:the expression \(3i + 27\).This will be written in standard complex form: \(27 + 3i\).

Key Concepts

Distributive Property

When you face a mathematical expression that involves multiplying a single term with a set of terms inside parentheses, you use something called the distributive property. This property helps you simplify expressions. Essentially, it allows you to take a term outside the parentheses and multiply it by each term inside the parentheses separately.
For example, in the expression \[ -3i(-1 + 9i) \], we apply the distributive property like so:

• Multiply \( -3i \) by \( -1 \)
• Multiply \( -3i \) by \( 9i \)

By performing these operations, you break down the expression and make it simpler to understand and solve. It's like peeling back layers of a complex problem to make it manageable.

Imaginary Unit

The imaginary unit, noted as \(i\), is a fascinating concept in mathematics. It's defined by the property \(i^2 = -1\). This might seem strange because normally, when we square a number, we get a positive result. However, the imaginary unit opens up a whole new world of numbers beyond the usual set of real numbers.
In complex number operations, the imaginary unit often appears in multiplication scenarios:
When you multiply \(i\) by itself, you get \[ i^2 = -1 \]. Therefore, in the problem \((-3i) \times (9i)\), we use this property:

• First, multiply the coefficients: \(-3 \times 9 = -27\)
• Multiply the imaginary parts: \(i \times i = i^2\)
• Since \(i^2 = -1\), we have \(-27 \times -1 = 27\)

Understanding \(i\)'s unique characteristics helps in managing operations that involve imaginary numbers.

Complex Form

Complex numbers are numbers that have two parts: a real part and an imaginary part. They are written in the form \(a + bi\) where \(a\) is the real part and \(b\) is the coefficient of the imaginary part.
This form makes them easy to manipulate, as seen in the original exercise. After using the distributive property and the characteristics of the imaginary unit, we get an expression like \(3i + 27\).
To write this in the standard complex form, it helps to keep the real part first: \[ 27 + 3i \]. This convention aids in organizing complex numbers consistently, making calculations, comparisons, and interpretations straightforward.

• Rearrange parts if needed, always with the real term first
• This format is universally adopted in complex number calculations

This systematic organization into a standard form is central in the study of complex numbers.