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Chapter 7: Problem 51

Multiply and simplify. Write each answer in the form \(a+b i\). $$(6-5 i)(3+4 i)$$

Short Answer

Expert verified
The answer is \( 38 + 9i \).

Step by step solution

01

- Apply the distributive property

Use the distributive property (also known as FOIL) to expand the expression \( (6 - 5i)(3 + 4i) \). This means multiplying each term in the first binomial by each term in the second binomial: \[ 6 \cdot 3 + 6 \cdot 4i - 5i \cdot 3 - 5i \cdot 4i \].
02

- Multiply the terms

Compute each multiplication: \[ 6 \cdot 3 = 18 \] \[ 6 \cdot 4i = 24i \] \[ -5i \cdot 3 = -15i \] \[ -5i \cdot 4i = -20i^2 \].
03

- Simplify using the property of imaginary units

Recall that \(i^2 = -1\), so the term \(-20i^2\) becomes \(-20(-1) = 20\).
04

- Combine the real and imaginary parts

Combine the results from the multiplications and simplifications: \[ 18 + 24i - 15i + 20 \] which simplifies to \[ (18 + 20) + (24i - 15i) = 38 + 9i \].

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Key Concepts

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

distributive property
The distributive property is a key algebraic principle used in mathematics. It allows you to multiply a sum by distributing the multiplication to each term inside the parentheses.
In the context of complex numbers, the distributive property helps expand expressions like \((6 - 5i)(3 + 4i)\).
Here's how it works:
  • You take each term in the first binomial (6 and -5i) and multiply it by each term in the second binomial (3 and 4i).
  • So you perform four multiplications: 6路3, 6路4i, -5i路3, and -5i路4i.
This gives: \[ 6 \times 3 + 6 \times 4i - 5i \times 3 - 5i \times 4i \]
Understanding this property is crucial because it provides the foundation for combining and simplifying terms in complex number multiplication.
FOIL method
The FOIL method stands for First, Outside, Inside, and Last.
It's a specific way to apply the distributive property to multiply two binomials.
Here's how the FOIL method is used to expand \((6 - 5i)(3 + 4i)\):
  • First: Multiply the first terms in each binomial: 6路3 = 18.
  • Outside: Multiply the outer terms in the binomials: 6路4i = 24i.
  • Inside: Multiply the inner terms: -5i路3 = -15i.
  • Last: Multiply the last terms in each binomial: -5i路4i = -20i^2.
Combining these, you get: \[18 + 24i - 15i - 20i^2 \] Next, you simplify using the properties of imaginary units.
This method is beneficial because it breaks the multiplication process into manageable steps.
imaginary unit
In complex number operations, the imaginary unit (denoted as \(i\)) is essential. By definition, \(i\) is the square root of -1:
\[ i = \sqrt{-1} \] Consequently, \(i^2 = -1\).
When multiplying complex numbers, terms involving \(i^2\) appear frequently.
For instance, in the expression \[ (6 - 5i)(3 + 4i) \], one term is -20i^2.
To simplify, use the property \(i^2 = -1\): \[-20i^2 = -20 (-1) = 20 \]
This conversion is crucial for combining terms accurately. Understanding the imaginary unit and its properties helps manage and simplify the results of complex number multiplication effectively.

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