Problem 82 - Intermediate Algebra

Question

Find each of the products and express the answers in the standard form of a complex number. $$ (5-7 i)(5+7 i) $$

Step-by-Step Solution

Answer

The product is 74.

Step 1: Apply the Distributive Property

To find the product of two complex numbers, we start by applying the distributive property (FOIL method). This means multiplying each part of the first complex number by each part of the second complex number.First: \(5 \times 5 = 25\)Outside: \(5 \times 7i = 35i\)Inside: \(-7i \times 5 = -35i\)Last: \(-7i \times 7i = -49i^2\)

Step 2: Simplify the Imaginary Terms

Now, simplify by combining like terms. Notice that the imaginary units will cancel out:\(35i - 35i = 0\)And simplify \(-49i^2\) using the fact that \(i^2 = -1\):\(-49i^2 = -49(-1) = 49\)

Step 3: Combine the Real Parts

Now add the results from the real parts and the simplified imaginary term:\(25 + 49 = 74\)

Step 4: Express in Standard Form

The standard form of a complex number is expressed as \(a + bi\). Since the imaginary part has canceled out, our product is:\(74 + 0i\), which simplifies to \(74\).

Key Concepts

Standard Form

The standard form of a complex number is a structured way to express complex numbers. In mathematics, every complex number can be written in the form \(a + bi\), where:

• \(a\) is the real part of the complex number.
• \(b\) is the imaginary part coefficient.
• \(i\) is the imaginary unit, which satisfies \(i^2 = -1\).

When presenting a complex number, it's important to distinguish between the real and imaginary components clearly. For instance, if a problem results in a number like \(74 + 0i\), it simplifies to \(74\), since \(0i\) indicates there is no imaginary component. Expressing the result in standard form helps in easily identifying and working with real and imaginary parts. This form becomes especially handy in arithmetic operations, such as addition, subtraction, and multiplication of complex numbers.

Distributive Property

The distributive property is a fundamental algebraic principle used frequently in mathematics. It allows us to multiply a single term across terms in a parenthesis. For expressions like \((a + b)(c + d)\), the distributive property helps break it down so that every term in the first bracket is multiplied by every term in the second.
In the context of complex numbers, using the distributive property becomes necessary when dealing with products like \((5 - 7i)(5 + 7i)\). Each component is distributed and multiplied:

• First: \(5\times5\)
• Outside: \(5\times7i\)
• Inside: \(-7i\times5\)
• Last: \(-7i\times7i\)

This multi-step process is also known as the FOIL method, which is a mnemonic for ensuring each pair of terms is accounted for in the multiplication.

FOIL Method

The FOIL method is a specific application of the distributive property. It stands for First, Outside, Inside, Last, and guides us in multiplying two binomials.
In our example, \((5 - 7i)(5 + 7i)\), FOIL helps in organizing the multiplication process:

• First: Multiply the first terms (\(5 \times 5\)) resulting in 25.
• Outside: Multiply the outer terms (\(5 \times 7i\)) yielding \(35i\).
• Inside: Multiply the inner terms (\(-7i \times 5\)) to get \(-35i\).
• Last: Multiply the last terms (\(-7i \times 7i\)) which results in \(-49i^2\).

This process ensures all parts are multiplied consistently, which is especially important in complex number multiplication where real and imaginary parts intermingle.

Imaginary Units

Imaginary units are a cornerstone of complex numbers. They introduce the concept of \(i\), where \(i^2 = -1\), which seems counterintuitive at first because squaring a number usually results in a positive. The use of this unit allows us to express numbers that would otherwise be impossible to define using just real numbers.
In the solution to \((5 - 7i)(5 + 7i)\), understanding \(i^2\) is crucial. We multiply the imaginary parts, \(-7i \times 7i\), resulting in \(-49i^2\). Due to \(i^2 = -1\), this becomes \(-49(-1)\), simplifying to 49. Hence, the imaginary components cancel each other out, illustrating how imaginary units can greatly influence the final result of complex number operations. This unique property makes complex numbers a versatile mathematical tool for solving a wide range of problems.