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Chapter 8: Problem 74

Assume \(x\) represents a real number and multiply \((x-4 i)(x+4 i)\).

Short Answer

Expert verified
The product is \(x^2 + 16\).

Step by step solution

01

Identify the Expression

The given problem involves multiplying two complex conjugates: \((x-4i)\) and \((x+4i)\). This expression takes the form \((a-b)(a+b)\) where \(a = x\) and \(b = 4i\).
02

Apply the Difference of Squares Formula

Use the difference of squares formula, which states \((a-b)(a+b) = a^2 - b^2\). Here, substitute \(a = x\) and \(b = 4i\) to get \((x)^2 - (4i)^2\).
03

Compute the Squares

Calculate \(x^2\) and \((4i)^2\): - \(x^2\) is simply \(x^2\).- \((4i)^2 = 16i^2\). Since \(i^2 = -1\), it follows that \(16i^2 = 16(-1) = -16\).
04

Simplify the Expression

Combine the results from Step 3 to get: \(x^2 - 16i^2 = x^2 - (-16) = x^2 + 16\). This is the product of the original expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Conjugates
Complex conjugates are a foundational concept in complex numbers. Simply put, a complex number is composed of a real part and an imaginary part. For example, if you have a complex number such as \[ a + bi \] (where \(i\) is the imaginary unit), its complex conjugate is \\( a - bi \). \This mathematical tool is especially useful when simplifying expressions involving complex numbers.

In the exercise problem, \((x-4i)\) and \((x+4i)\) are complex conjugates.

Notice that the only difference is the sign in front of \(4i\).

This property makes them ideal for applying the difference of squares formula, as it helps in removing the imaginary part and simplifies calculations.
Difference of Squares
The difference of squares is a useful algebraic identity that simplifies expressions into more manageable parts. The formula is written as: \[(a-b)(a+b) = a^2 - b^2\]

This formula is especially handy because it allows you to eliminate middle terms by taking advantage of the opposite signs, resulting in two opposing terms that cancel each other out.

In the exercise, substituting \(a = x\) and \(b = 4i\) into the formula results in: \\[ (x-4i)(x+4i) = x^2 - (4i)^2 \]

This sets the stage for further simplification, as recognizing and applying this identity helps reduce the expression smoothly.
Imaginary Unit
The imaginary unit, denoted by \(i\), is an essential concept when dealing with complex numbers. Its defining property is that \(i^2 = -1\).

Understanding this allows us to simplify expressions involving imaginary numbers. For example, when squaring the imaginary component \((4i)\) in the given exercise, we calculate: \((4i)^2 = 16i^2\).

Knowing that \\(i^2 = -1\), we replace \(16i^2\) with \(16(-1) = -16\).

This substitution effectively transforms a complex product into a real number, which is essential for simplifying expressions in problems involving complex conjugates.

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Most popular questions from this chapter

Use the formula for \(\cos (A+B)\) to find the exact value of \(\cos 75^{\circ}\).

Use a calculator to help write each complex number in standard form. Round the numbers in your answers to the nearest hundredth.\(10 \operatorname{cis} 6\)

Let \(z_{1}=2+3 i, z_{2}=2-3 i\), and \(z_{3}=4+5 i\), and find \(z_{1} z_{2}\)

Show that 2 cis \(60^{\circ}\) and 2 cis \(\left(-60^{\circ}\right)\) are conjugates.

Show that if \(z=\cos \theta+i \sin \theta\), then \(|z|=1\).
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