Problem 16 - College Algebra Essentials

Question

In Exercises \(9-20,\) find each product and write the result in standard form. $$(2+7 i)(2-7 i)$$

Step-by-Step Solution

Answer

So, the product of (2+7i)(2-7i) in standard form is 53.

Step 1: Write binomial multiplication

The product (2+7i)(2-7i) breaks down following the distributive property something like: (a+b)(c-d) which expands into ac - ad + bc - bd.

Step 2: Substitute given values into the formula

Substitute a=2, b=7i, c=2 and d=7i into the formula from Step 1: 2*2 - 2*7i + 7i*2 - 7i*7i.

Step 3: Multiply and simplify

Multiply the terms and simplify them, remember that the multiplication of two imaginary numbers \(i * i\) gives -1, so \(7i*7i= -49\). When substituting and multiplying, we get: 4 - 14i + 14i - (-49) = 4 + 49 = 53.

Key Concepts

Binomial Multiplication

Binomial multiplication involves multiplying two binomials, which are algebraic expressions containing two terms. In the exercise provided, our binomials are

• First Binomial: \( 2 + 7i \)
• Second Binomial: \( 2 - 7i \)

Binomial multiplication is quite common in algebra, especially when dealing with complex numbers. Using the pattern \( (a + b)(c + d) \), we combine and expand to obtain a single expression. This process usually employs the distributive property multiple times to ensure that every term in the first binomial is multiplied by every term in the second. This expansion is essential to convert a product into a simpler expression, which can then be simplified further.

Distributive Property

The distributive property is a fundamental principle in mathematics that allows us to multiply a term across the terms inside a parenthesis. In the context of binomial multiplication, it tells us how to properly expand the product of two binomials.
The key idea is to distribute each term in one binomial to each term in the other binomial. For example, for our expression \((2 + 7i)(2 - 7i)\), we break it down as:

• \(2 \times 2 \)
• \(2 \times (-7i)\)
• \(7i \times 2\)
• \(7i \times (-7i) \)

This step-by-step application of the distributive property is what allows us to expand and simplify complex expressions accurately.

Imaginary Unit

The imaginary unit, denoted as \(i\), is an essential component when working with complex numbers. It is defined by the equation \(i^2 = -1\).
In the exercise, the term \(7i \times 7i\) results in \(49i^2\).
However, using the definition of the imaginary unit, we know \(i^2 = -1\),
thus \(49i^2 = 49(-1) = -49\).
Recognizing how to utilize \(i\) is vital in simplifying expressions with complex numbers. It enables us to handle squared terms involving imaginary numbers. This understanding is critical, especially when reducing and simplifying complex products.

Standard Form

Standard form for complex numbers is \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.
The goal is to express any complex expression as a sum of a real number and an imaginary number.
In our exercise, starting with \(4 - 14i + 14i - (-49)\), we simplify to \(4 + 49\) by cancelling the imaginary terms \(-14i + 14i\). The result \(53\) falls under complex number's standard form with \(a = 53\) and \(b = 0\), where there is no imaginary part visible.