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Chapter 7: Problem 422

In the following exercises, simplify the given expression. $$ -10(0.4 n+0.7) $$

Short Answer

Expert verified
-4n - 7

Step by step solution

01

Distribute the Constant

First, distribute the -10 throughout the parentheses by multiplying -10 by each term inside the parentheses. This means multiplying -10 by 0.4n and -10 by 0.7.
02

Multiply the First Term

Multiply -10 by 0.4n to get -4n. The equation now looks like this: -4n.
03

Multiply the Second Term

Next, multiply -10 by 0.7 to get -7. Now the equation looks like this: -4n - 7.
04

Combine the Terms

Combine the terms to finalize the simplified expression. Thus, the simplified expression is -4n - 7.

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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 concept in algebra and prealgebra. It allows you to simplify expressions by distributing, or multiplying, a single term through a set of parentheses.
For instance, in the expression \[a(b + c) = ab + ac\], the term \[a\] is distributed and multiplied by both \[b\] and \[c\].
In our problem \[-10(0.4n + 0.7)\], we used the distributive property to multiply \[-10\] by \[0.4n\] and \[0.7\]. This gives us \[-10 \times 0.4n + (-10) \times 0.7\].
It's a handy tool to break expressions into simpler parts.
Multiplication of Constants
Multiplying constants involves straightforward arithmetic but often requires attention to detail. In the exercise, we had to multiply \[-10\] by each term within the parentheses separately.
First, we multiplied \[-10 \times 0.4n\], which gave us \[-4n\]. We then multiplied \[-10 \times 0.7\], resulting in \[-7\].
Working through these steps carefully ensures accuracy and reinforces your understanding of basic arithmetic with negative numbers. Remember, multiplying by a negative number alters the sign of the product.
Combining Like Terms
Combining like terms is an essential skill in algebra for simplifying expressions. Like terms are terms that have the same variable raised to the same power.
In our simplified expression \[-4n - 7\], we combined \[-4n\] (a term with \[n\]) and \[-7\] (a constant term). Since they are unlike terms, they cannot be further combined.
Combining like terms streamlines expressions and solves them in fewer steps, making solving equations easier.
Prealgebra
Prealgebra prepares you for more advanced algebra concepts by focusing on arithmetic involving variables, constants, and basic operations.
In the exercise, we applied core prealgebra concepts like the distributive property, multiplication of constants, and combining like terms.
Understanding these principles lays the foundation for solving more complex algebraic equations and is crucial for progressing in mathematics. Prealgebra ensures you grasp these basics thoroughly.

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