Problem 66 - Algebra for College Students

Question

Find each of the products and express the answers in the standard form of a complex number. $$(7 i)(-9+3 i)$$

Step-by-Step Solution

Answer

The product is \(-21 - 63i\) in standard form.

Step 1: Apply the Distributive Property

To find the product of two complex numbers, apply the distributive property. Multiply each term inside the parentheses by the number outside: \((7i)(-9) + (7i)(3i)\).

Step 2: Multiply Each Term

First, calculate \( (7i)(-9) = -63i \). Next, calculate \((7i)(3i) = 21i^2 \). Recall that \(i^2 = -1\), so \(21i^2 = 21(-1) = -21\).

Step 3: Simplify and Combine Terms

Combine the real and imaginary parts: The real part is \(-21\), and the imaginary part is \(-63i\). Thus, the product is \(-21 - 63i\).

Step 4: Express in Standard Form

The standard form for a complex number is \(a + bi\). The result \(-21 - 63i\) is already in standard form with \(a = -21\) and \(b = -63\).

Key Concepts

Distributive Property

The distributive property is a fundamental principle used in mathematics to simplify expressions and calculations. It enables us to multiply a single term by each term inside of a parenthesis separately. In the context of complex numbers, like in our exercise

• Distributive property is applied when multiplying complex numbers or terms.
• It's essential to ensure that each part of the expression gets multiplied by the external term.
• For example, when applying the distributive property to (\(7i\)(-9 + 3i)), you perform the calculations: (\(7i\)(-9) + \(7i\)(3i)).

This property allows us to handle the multiplication efficiently by breaking it down into manageable pieces, ensuring we don't skip any part of the expression.

Standard Form of a Complex Number

The concept of the standard form of a complex number is straightforward yet crucial when expressing the results of complex number operations. Every complex number is written in the form \(a + bi\), where

• \(a\) represents the real part, and
• \(b\) represents the imaginary part multiplied by the imaginary unit \(i\).

In some exercises or problems, after performing calculations, you might need to rearrange or simplify your results to fit this form. In the solution of our given exercise, we end up with \(-21 - 63i\), which is already in the standard form where \(a = -21\) and \(b = -63\). This format helps us easily identify and work with the real and imaginary components of a complex number.

Imaginary Unit (i)

The imaginary unit, denoted as \(i\), is a unique component of complex numbers with specific properties that help us perform calculations involving numbers that lie beyond the real number line. The defining property of \(i\) is:

• \(i^2 = -1\)

This property is fundamental as it differentiates complex numbers from real numbers. Every time \(i^2\) appears in calculations, it should immediately be replaced with \(-1\) to simplify expressions. For instance, in our exercise's solution, we had \(7i\times 3i = 21i^2\). Recognizing that \(i^2 = -1\) transforms this term into \(21 \times (-1) = -21\), showing how \(i\) can change the nature of calculations significantly. Understanding \(i\) is essential in working with and simplifying complex number operations.

Complex Number Multiplication

Multiplying complex numbers is an operation similar to multiplying binomials, where distributing each term is vital. In carrying out complex number multiplication, you can follow these steps:

• Apply the distributive property to handle each part separately.
• Multiply each pair of terms carefully, keeping track of the real and imaginary components.
• Simplify the result by combining like terms and utilizing the properties of \(i\), most notably \(i^2 = -1\).

For our exercise, multiplying \(7i\) by \(-9 + 3i\) involves first distributing \(7i\) to each component, resulting in the terms \(-63i\) and \(21i^2\). After simplifying \(21i^2 = -21\), we combine these to get the final result: \(-21 - 63i\), which is ultimately placed in the complex number's standard form. Mastery of these multiplication steps ensures accurate and efficient calculations with complex numbers.