Problem 66 - A Graphical Approach to Precalculus with Limits

Question

Multiply as indicated. Write each product in standand form. $$(\sqrt{2}-4 i)(\sqrt{2}+4 i)$$

Step-by-Step Solution

Answer

The product is 18.

Step 1: Identify the Expression

The problem is asking us to multiply two binomials: \[(\sqrt{2} - 4i)(\sqrt{2} + 4i)\]. This is a special product called the difference of squares, given by \((a - b)(a + b) = a^2 - b^2\). In this case, a = \sqrt{2}\ and b = 4i.

Step 2: Apply the Difference of Squares Formula

Use the formula for the difference of squares: \((a - b)(a + b) = a^2 - b^2\).Substitute \(a = \sqrt{2}\) and \(b = 4i\):a^2 = (\sqrt{2})^2 = 2.b^2 = (4i)^2 = 16i^2. Since \(i^2 = -1\), we get \(b^2 = 16(-1) = -16\).

Step 3: Simplify the Expression

Substitute the computed squares into the difference of squares formula:\(a^2 - b^2 = 2 - (-16)\).Simplify further: \(2 + 16 = 18\).

Step 4: Write the Product in Standard Form

After applying the difference of squares and simplifying, the product of the expression \((\sqrt{2} - 4i)(\sqrt{2} + 4i)\) is 18. Standard form for complex numbers is \(a + bi\), which in this case is simply \(18 + 0i\). This means our expression is just a real number: 18.

Key Concepts

Difference of squares

The difference of squares is a mathematical pattern that's particularly handy when dealing with algebraic expressions. It’s like having a secret shortcut for solving certain equations. When you see a pair of binomials in the form

• the first has a plus sign
• the second has a minus sign

it spells the difference of squares formula: \[(a - b)(a + b) = a^2 - b^2 \].This formula tells you that instead of multiplying everything out, you only need to square the first term and subtract the square of the second term. Pretty neat, right? Let's put it in context with our exercise. We identified

• \(a = \sqrt{2}\) and
• \(b = 4i\).

So, according to the formula, the product comes down to simply calculating: \[a^2 - b^2 = (\sqrt{2})^2 - (4i)^2 = 2 - (16 \times i^2)\]. Don't forget that the imaginary unit changes things up here!

Binomials

Binomials are like the simplest building blocks in algebra that aren't just plain numbers. A binomial is simply an expression that consists of two terms. For example, in the expression

• \((\sqrt{2} - 4i)\)

, you have two parts:

• \(\sqrt{2}\), which is a real number
• \(-4i\), which is an imaginary part.

Another characteristic of a binomial is that people like to perform operations on them, such as addition or multiplication, like what you did in this exercise. When multiplying two binomials, the process involves using each term in one binomial to multiply each term in the other. This might remind you of larger binomials products, but when they follow a pattern like the difference of squares, the work is simplified greatly!

Imaginary unit

The imaginary unit, denoted by \(i\), is a concept that extends our number system into something more creative. While regular numbers (integers, fractions, etc.) live on a straight number line, imaginary numbers take us into a whole new dimension, a perpendicular line if you will. Math talks about \(i\) as the square root of -1, a number you can't find among the regular real numbers because no real number squared will give you -1. If you wonder why this matters in our exercise, look at what happens when we square the imaginary part \((4i)^2\). Here, the square involves both the number (4) and \(i\):

• First, calculate \((4)^2 = 16 \).
• Then, consider \(i^2 = -1 \).

Combine them, and you get \(16 \times (-1) = -16 \). This change, thanks to the imaginary unit, flips the sign of the squared number and plays a key role in simplifying expressions just like ours into real numbers.