Problem 48 Multiply and simplify. $$(3-4 ... [FREE SOLUTION]

Chapter 11: Problem 48

Multiply and simplify. $$(3-4 i)(6+7 i)$$

Short Answer

Expert verified

The simplified product of the given complex numbers is $-10 - 3i$.

Step by step solution

Multiply the two complex numbers (FOIL method)

Using the FOIL method, we multiply the two complex numbers (3 - 4i) and (6 + 7i) as follows: $(3 - 4i)(6 + 7i) = (3)(6) + (3)(7i) + (-4i)(6) + (-4i)(7i)$

Calculate each multiplication term

Now, we will calculate each term: $ (3)(6) = 18 $ $ (3)(7i) = 21i $ $ (-4i)(6) = -24i $ $ (-4i)(7i) = 28i^2 $ Remember that $i^2 = -1$ since $i = \sqrt{-1} $. So, $ 28i^2 = 28(-1) = -28 $.

Combine the terms and simplify

Finally, we combine all the terms and simplify the expression: $(3 - 4i)(6 + 7i) = 18 + 21i - 24i - 28 = (18 - 28) + (21 - 24)i$ $(3 - 4i)(6 + 7i) = -10 - 3i$ Thus, the simplified product of the given complex numbers is $-10 - 3i$.

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Key Concepts

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

FOIL Method

The FOIL method is a technique used to multiply two binomials. It鈥檚 an acronym that stands for First, Outer, Inner, Last, which helps you remember the order in which to multiply the terms. When you have two complex numbers, like

(3 - 4i)
(6 + 7i)

using the FOIL method means:

First: Multiply the first terms in each binomial: 3 and 6.
Outer: Multiply the outer terms: 3 and 7i.
Inner: Multiply the inner terms: -4i and 6.
Last: Multiply the last terms in each binomial: -4i and 7i.

This method ensures that every term interacts with each other term in the binomials, resulting in four initial products that we then combine or simplify further.

Imaginary Unit

The imaginary unit, denoted as $i$, is defined as the square root of -1. In mathematics, imaginary numbers are used when negative numbers appear under the square root in equations. Here's what you need to remember:

The property $i^2 = -1$ is crucial. It allows us to simplify expressions where $i$ is squared.
When multiplying complex numbers, the appearance of $i^2$ lets you replace it with -1, altering the sign of those products.

For example, in the expression $(-4i)(7i)$, when you arrive at $28i^2$, remembering that $i^2 = -1$ turns this into -28. This step is key in simplifying the resulting expression when dealing with complex numbers.

Simplifying Expressions

After utilizing the FOIL method and applying the properties of the imaginary unit, the final crucial step is simplifying the expression. This involves combining like terms and reducing the expression to its simplest form. Let's break it down:

First, compute each product obtained from the FOIL method.
Next, rewrite the equation by combining real terms separately from the imaginary terms.
In our example: gather the real components 18 and -28 together, and the imaginary components 21i and -24i.

This means you'll sum up 18 and -28 for the real part, which yields -10. For the imaginary terms, combine 21i and -24i resulting in -3i. So, the product $(3 - 4i)(6 + 7i)$ finally simplifies to $-10 - 3i$. Each step in simplifying helps reveal the elegant property of complex numbers, represented in a clean and straightforward result.

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91影视

Multiply and simplify. $$(3-4 i)(6+7 i)$$

Short Answer

Step by step solution

Multiply the two complex numbers (FOIL method)

Calculate each multiplication term

Combine the terms and simplify

Key Concepts

FOIL Method

Imaginary Unit

Simplifying Expressions

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