Problem 40 Perform the operation and write ... [FREE SOLUTION]

91影视

Elementary Algebra Within Reach

Ron Larson, Kimberly Nolting

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 10: Problem 40

Perform the operation and write the result in standard form. $$(9-12 i)(15+18 i)$$

Short Answer

Expert verified

The product of the complex numbers is $ -81 + 342i $

Step by step solution

Distribute Multiplication

Firstly, distribute multiplication as you would with binomials. So, calculate $ (9 \times 15 - 12 \times 18) $ for the real part and $ (9 \times 18 + 12 \times 15) $ for the imaginary part.

Multiply Imaginary Parts

Now, the real part of the product is given by $ 9 \times 15 - 12 \times 18 = 135 - 216 = -81 $. For the imaginary part, compute $ 9 \times 18 + 12 \times 15 = 162 + 180 = 342 $. Therefore, before simplification, the product is $ -81 + 342i $

Write in Standard Form

Finally, we recognize that $ -81 + 342i $ is already in standard form (real + imaginary), and therefore, is the final result.

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Key Concepts

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

Understanding Binomials in Complex Numbers

Binomials are expressions with two terms. In complex numbers, these terms often involve a real part and an imaginary part. For instance, in the expression

$(9 - 12i)$
$(15 + 18i)$

we treat this like the multiplication of regular binomials.
When multiplying these binomials, we apply the distributive property, which is often called FOIL (First, Outer, Inner, Last) in mathematics. It鈥檚 a strategy for organizing the multiplication of two binomials:

First: Multiply the first terms of each binomial.
Outer: Multiply the outer terms of the binomial.
Inner: Multiply the inner terms of the binomial.
Last: Multiply the last terms of each binomial.

So, you multiply as
$(9 \times 15) - (12 \times 18i) + (9 \times 18i) + (12i \times 15i)$.

Understanding this will help effectively break down multiplication involving complex numbers.

Converting to Standard Form

Standard form for complex numbers is expressed as $a + bi$, where $a$ is the real part and $bi$ is the imaginary part. After calculating the multiplication of complex binomials, simplifying into standard form is a check to ensure clarity and consistency in your results.
In our process:

We derived a compound expression $-81 + 342i$ after our multiplications.
This is already in the standard form $-81$ (real part) and $342i$ (imaginary part).

It鈥檚 crucial to verify that the resulting terms are correctly placed. The real number is the "a" value, and the imaginary coefficient is the "b" value in $a + bi$.
This provides a structured way to present complex numbers that is easily interpretable.

Imaginary Numbers Explained

Imaginary numbers are components of complex numbers, represented by $i$. Here, $i$ is defined as $\sqrt{-1}$. When squared, $i^2 = -1$.
Imaginary numbers play a critical role in solving equations that might not have solutions in the real number system.
In operations with complex numbers:

The imaginary part is what involves $i$.
It鈥檚 treated as a variable, with specific rules coming into play, like $i^2 = -1$.

For the exercise given:

The real part, like $-81$, doesn't involve $i$.
The imaginary part, $342i$, stems from terms that multiply with $i$.

Mastering imaginary numbers enriches your understanding of complex numbers overall. They are fundamental in advanced mathematics and have applications in fields such as engineering and physics.

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91影视

Perform the operation and write the result in standard form. $$(9-12 i)(15+18 i)$$

Short Answer

Step by step solution

Distribute Multiplication

Multiply Imaginary Parts

Write in Standard Form

Key Concepts

Understanding Binomials in Complex Numbers

Converting to Standard Form

Imaginary Numbers Explained

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