Problem 64 - A Graphical Approach to College Algebra

Question

Multiply as indicated. Write each product in standard form. $$(6-4 i)(6+4 i)$$

Step-by-Step Solution

Answer

The product is 52.

Step 1: Identify the Expression Type

The expression to be solved is the product of two conjugates: \((6 - 4i)\) and \((6 + 4i)\). The conjugate pairs are of the form \((a - bi)(a + bi)\).

Step 2: Apply the Conjugates Property

Recall that the product of conjugates \((a - bi)(a + bi) = a^2 + b^2\). In this case, identify \(a = 6\) and \(b = 4\).

Step 3: Square and Sum the Real and Imaginary Components

Calculate \(a^2\) and \(b^2\): \[ 6^2 = 36 \quad \text{and} \quad 4^2 = 16 \]Add the two results: \[ 36 + 16 = 52 \]

Step 4: Express the Product in Standard Form

Since the result from multiplying conjugates is already a real number, the standard form of the product is simply the sum calculated in the previous step. Thus, the product in standard form is \(52\).

Key Concepts

Conjugates in Complex Numbers

In the realm of complex numbers, conjugates play a pivotal role. Imagine you have a complex number like \(a + bi\). Its conjugate is simply \(a - bi\). This means you keep the real part the same but swap the sign of the imaginary part. Conjugates are particularly helpful when you want to multiply complex numbers or rationalize their denominators.

When multiplying conjugate pairs, such as \((a + bi)(a - bi)\), the imaginary parts cancel each other out. The resulting product is the sum of the squares of the real and imaginary parts. This is represented as \(a^2 + b^2\), which is always a real number.

Key Reasons to Use Conjugates:

• They help simplify division of complex numbers.
• Multiplying conjugates eliminates imaginary parts, making calculations easier.
• They allow expression of results in a real number form, providing simplicity and clarity.

Standard Form of Complex Numbers

Standard form is a way to express complex numbers neatly and consistently, as \(a + bi\). Here, \(a\) represents the real component, and \(bi\) is the imaginary component. It's crucial because it provides a clear way to communicate and work with complex numbers.

Understanding the Format:

• The real part, \(a\), is a standard real number.
• The imaginary part is represented by \(bi\), where \(b\) is real, and \(i\) is the imaginary unit.

After performing operations like addition, subtraction, or multiplication, you'll often be asked to express your answer in standard form. This ensures clarity and consistency in communicating mathematical ideas.

For example, when we multiplied the conjugates \((6 - 4i)\) and \((6 + 4i)\), the outcome was a real number, 52. Since there's no imaginary part, 52 is already in its standard form.

Understanding the Imaginary Unit

The imaginary unit, represented as \(i\), is fundamental in working with complex numbers. It’s defined by the property \(i^2 = -1\). This definition allows us to extend the real number system and solve equations that lack real solutions.

Properties of the Imaginary Unit:

• When \(i\) is squared, you get \(-1\).
• The powers of \(i\) cycle every four terms, leading to interesting results like \(i^4 = 1\).
• It enables the construction of complex numbers, paving the way for broader mathematical applications.

When dealing with complex numbers, \(i\) holds the imaginary part and signifies that the part it accompanies is distinct from real numbers. In the product example, terms involving \(i\) disappeared in the multiplication of conjugates, thanks to how \(i\) behaves.