Problem 9 In Problems 1-26, write the give... [FREE SOLUTION]

Chapter 17: Problem 9

In Problems 1-26, write the given number in the form $a+i b$. $$ (2-3 i)(4+i) $$

Short Answer

Expert verified

The expression simplifies to $11 - 10i$.

Step by step solution

Distribute the Terms

To solve $(2 - 3i)(4 + i)$, we need to apply the distributive property (also known as the FOIL method for binomials). Multiply each term in the first complex number $(2 - 3i)$ by each term in the second complex number $(4 + i)$. This gives: \[ (2 \cdot 4) + (2 \cdot i) + (-3i \cdot 4) + (-3i \cdot i) \].

Simplify the Real Parts

Calculate the real parts from the products: $2 \cdot 4 = 8$, and the imaginary part product that contributes to the real part: $-3i \cdot i = 3$ (since $i^2 = -1$, $-3i \cdot i = -3(-1) = 3$). So, the total real part is $8 + 3 = 11$.

Simplify the Imaginary Parts

Next, calculate the imaginary parts: $2 \cdot i = 2i$ and $-3i \cdot 4 = -12i$. Combine these to get the imaginary part: $2i - 12i = -10i$.

Write in Standard Form

Combine the real and imaginary parts from Steps 2 and 3 to write the number in the form $a + ib$. We have the real part $11$ and the imaginary part $-10i$, giving us a final result of $11 - 10i$.

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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 fundamental algebraic rule. It allows us to multiply a single term by each term inside a parenthesis. This property can be particularly useful when multiplying binomials. In our example, the expression

$(2 - 3i)(4 + i)$

is expanded using the distributive property. We multiply each term in the first bracket by each term in the second bracket:

$2 \cdot 4$
$2 \cdot i$
$-3i \cdot 4$
$-3i \cdot i$

By carrying out these operations step-by-step, we can effectively simplify and solve expressions involving complex numbers.

Binomials

Binomials are algebraic expressions that contain two distinct terms. In our problem, both factors,

$(2 - 3i)$ and
$(4 + i)$

are binomials. When multiplying such expressions, the FOIL method, an application of the distributive property, is often used. FOIL stands for

First
Outside
Inside
Last

These represent the order of operations for multiplying each term in one binomial by each term in the other. This method allows us to break down what might initially be a complex operation into smaller, manageable steps.

Imaginary Unit

The imaginary unit, denoted as $i$, is a key concept in complex numbers. It is defined by the equation $i^2 = -1$. This definition allows us to represent and calculate expressions involving the square root of negative numbers. In our example

$-3i \cdot i = 3$

we see that when

$-3i$ is multiplied by $i$

the result is a real number. This is because $i^2$ is equal to

$-1$.

Understanding the imaginary unit is fundamental to working with complex numbers, simplifying expressions, and achieving results in standard form.

Standard Form of a Complex Number

The standard form of a complex number is expressed as $a + bi$. This format clearly separates the real part $a$ from the imaginary part $bi$. In our solved problem, we deduced that:

real part is 11
imaginary part is -10i

This combination gives us the complex number:

$11 - 10i$

By writing complex numbers in standard form, we can more easily add, subtract, multiply, or divide them, and easily recognize and apply operations on each component separately. It provides clarity and structure, enabling efficient calculations and communication of complex ideas.

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91影视

In Problems 1-26, write the given number in the form \(a+i b\). $$ (2-3 i)(4+i) $$

Short Answer

Step by step solution

Distribute the Terms

Simplify the Real Parts

Simplify the Imaginary Parts

Write in Standard Form

Key Concepts

Distributive Property

Binomials

Imaginary Unit

Standard Form of a Complex Number

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