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Chapter 3: Problem 63

Multiply as indicated. Write each product in standard form. $$(2+3 i)(2-3 i)$$

Short Answer

Expert verified
The product is -5.

Step by step solution

01

Recognize the Pattern

The expression \((2+3i)(2-3i)\) is a product of two conjugates. In general, \((a+bi)(a-bi)\) equals \(a^2+b^2i^2\).
02

Multiply the Conjugates

Using the formula for multiplying conjugates: \((2)^2 + (3i)^2 = 4 + 9(-1)\).
03

Simplify the Expression

Calculate to obtain \(4 - 9 = -5\).
04

Write in Standard Form

The standard form of a complex number is \(a + bi\), where \(b = 0\) in this case. Thus, the final result is \(-5\).

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Key Concepts

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

Conjugates of Complex Numbers
Conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. Understanding conjugates is key when working with complex numbers, especially in multiplication and division.
  • Definition: If you have a complex number, for example, \( a + bi \), its conjugate is \( a - bi \).
  • Importance: The product of a complex number and its conjugate always results in a real number. This is because the imaginary parts cancel each other out.
  • Example: Consider \( (2 + 3i) \) and its conjugate \( (2 - 3i) \). Their product is \( (2)^2 + (3i)^2 = 4 + 9(-1) = 4 - 9 = -5 \), which is a real number.
Applying conjugates simplifies complex number calculations and is especially useful in rationalizing denominators.
Standard Form of Complex Numbers
The standard form of a complex number is expressed as \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part.
  • Simplified Understanding: Think of it as a way to write complex numbers that highlights both their real and imaginary components.
  • Why It Matters: Writing complex numbers in standard form makes them easier to work with in operations like addition, subtraction, and multiplication.
  • Example: After performing the steps of multiplication or addition, ensure the result is in standard form. For the expression \( (2 + 3i)(2 - 3i) \), after simplifying, you get \( -5 + 0i \) or simply \( -5 \), which is still considered a valid standard form with no imaginary component.
Using standard form is crucial for recognizing the distinct contributions of the real and imaginary parts.
The Imaginary Unit (i)
The imaginary unit, denoted as \( i \), is a fundamental component in complex numbers. It's defined by the property that \( i^2 = -1 \).
  • Definition: \( i \) is used to represent the square root of -1, solving equations that don't have real solutions.
  • Role in Complex Numbers: A complex number generally looks like \( a + bi \), where \( b \) is multiplied by \( i \), indicating the imaginary part.
  • Example in Use: In the problem expression \( (2 + 3i)(2 - 3i) \), the term \( (3i)^2 \) evaluates to 9\( i^2 \). Given that \( i^2 = -1 \), this becomes \( 9(-1) = -9 \).
Knowing \( i \) and its properties is essential for understanding complex numbers as it allows extending real-number concepts into new dimensions of mathematical exploration.

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Most popular questions from this chapter

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$s=\frac{1}{2} g t^{2} \quad \text { for } t$$

Use Descartes' rule of signs to determine the possible numbers of positive and negative real zeros for \(P(x) .\) Then, use a graph to determine the actual numbers of positive and negative real zeros. $$P(x)=x^{5}+3 x^{4}-x^{3}+2 x+3$$

Use the boundedness theorem to show that the real zeros of \(P(x)\) satisfy the given conditions. \(P(x)=3 x^{4}+2 x^{3}-4 x^{2}+x-1\); no real zero less than \(-2\)

Divide. $$\left(x^{2}+\frac{1}{2} x-1\right) \div(2 x+1)$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$\mathscr{A}=\pi r^{2} \quad \text { for } r$$
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