Problem 75 Let $z_{1}=2+3 i, z_{2}=2-3 i$... [FREE SOLUTION]

91影视

Trigonometry

Charles P. McKeague, Mark D. Turner

$Math Studyset 91影视 Explanations$ Math

8 Edition

Chapter 8: Problem 75

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

Short Answer

Expert verified

The product $ z_{1}z_{2} $ is 13.

Step by step solution

Write down the expression for multiplication

We need to find the product of two complex numbers: $ z_{1} = 2 + 3i $ and $ z_{2} = 2 - 3i $. This expression can be written as $ z_{1}z_{2} = (2 + 3i)(2 - 3i) $.

Apply the distributive property (FOIL Method)

When multiplying two complex numbers, use the distributive property: \[(2 + 3i)(2 - 3i) = 2(2) + 2(-3i) + 3i(2) + 3i(-3i)\]This simplifies to:\[= 4 - 6i + 6i - 9i^2\]

Simplify the expression

Notice that the $-6i$ and $+6i$ terms cancel each other out. We then have:\[= 4 - 9i^2\]Recall that $i^2 = -1$, substitute this into the equation:\[= 4 - 9(-1)\]$= 4 + 9$

Final simplification

Now, we add the real numbers:$= 4 + 9 = 13$Hence, the product $ z_{1} z_{2} $ simplifies to 13.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

FOIL Method

When multiplying two binomials, such as complex numbers, the FOIL method is a handy technique. FOIL stands for First, Outer, Inner, and Last 鈥� these are the pairs of terms we multiply together to get all parts of the product.
In the case of complex numbers, like

$ (a + bi) $ and $ (c + di) $

the process looks like this:

First: multiply the first terms of each binomial ($ a \times c $).
Outer: multiply the outer terms ($ a \times di $).
Inner: multiply the inner terms ($ bi \times c $).
Last: multiply the last terms of each binomial ($ bi \times di $).

For the example of $ (2+3i)(2-3i) $, we apply the FOIL process to multiply:

First: $ 2 \times 2 = 4 $
Outer: $ 2 \times (-3i) = -6i $
Inner: $ 3i \times 2 = 6i $
Last: $ 3i \times (-3i) = -9i^2 $

The FOIL method helps in ensuring no term is missed when expanding the product.

Distributive Property

The distributive property is a key mathematical principle, especially in algebra, and it's vital for multiplying expressions like complex numbers. The property states:

$ a(b+c) = ab + ac $

When it comes to complex numbers, this property helps organize and distribute the multiplication across each term.
In the problem of multiplying

$ (2 + 3i) \times (2 - 3i) $

we break it down:

First, multiply each term in the first complex number by each term in the second, then add the results together.

By following the structured distribution:

$ (2 \times 2) + (2 \times -3i) + (3i \times 2) + (3i \times -3i) $

All terms are integrated into one finished result:

$ 4 - 6i + 6i - 9i^2 $

This method ensures that no part is left out in the multiplication.

Imaginary Unit i

The imaginary unit $ i $ is an essential concept in complex numbers. It represents the square root of $ -1 $ and is written as:

$ i^2 = -1 $

In expressions involving complex numbers, $ i $ functions like a regular variable, but follows different rules due to its definition.
For example, when dealing with the expression

$ 3i \times -3i = -9i^2 $

we simplify it using the rule for $ i^2 $. Since $ i^2 = -1 $, substitute it to get:

$ -9 \times -1 = 9 $

By understanding and correctly applying the properties of $ i $, it makes simplifying and working with complex numbers possible. So, in operations like multiplication, always recall $ i^2 = -1 $ and use it to transform the expression.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

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

Short Answer

Step by step solution

Write down the expression for multiplication

Apply the distributive property (FOIL Method)

Simplify the expression

Final simplification

Key Concepts

FOIL Method

Distributive Property

Imaginary Unit i

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Theoretical and Mathematical Physics

Mechanics Maths

Decision Maths

Calculus

Study anywhere. Anytime. Across all devices.