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Chapter 1: Problem 42

$$\text { Simplify the given algebraic expressions.}$$ $$-2 F+2((2 F-1)-5)$$

Short Answer

Expert verified
The simplified expression is \(2F - 12\).

Step by step solution

01

Distribute through the Parentheses

Begin by distributing the 2 into the expression inside the first parenthesis: \[ 2 ((2F - 1) - 5) = 2(2F - 1) + 2(-5) \] Simplifying these terms:\[ = 4F - 2 - 10 \] Combine like terms:\[ = 4F - 12 \]
02

Combine with the Initial Term

Now take the expression you found, \(4F - 12\), and combine it with \(-2F\) which was outside the parentheses initially:\[ -2F + (4F - 12) \]
03

Simplify the Expression

Combine like terms by adding \(-2F\) to \(4F\):\[ (-2F + 4F) - 12 = 2F - 12 \] Thus, the simplified expression is:\[ 2F - 12 \]

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Key Concepts

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

Understanding the Distributive Property
The distributive property is a fundamental tool when working with algebraic expressions. It allows you to multiply a single term by each term within a set of parentheses. In our exercise, we began by distributing 2 across the expression inside the parentheses:\[ 2((2F - 1) - 5) = 2(2F - 1) + 2(-5) \]
Here's a simple breakdown:
  • Take the number outside, which is 2 in this case, and multiply it by each term inside the parentheses.
  • This gives us two operations: one for \(2(2F - 1)\) and another for \(2(-5)\).
This step eliminates the parentheses and lets us work with each term directly. We end up with:\[ 4F - 2 - 10 \]
This step is crucial because it sets the stage for combining like terms, which is the next important operation in simplifying expressions.
Combining Like Terms in Algebraic Expressions
Combining like terms is all about simplifying your expression by adding or subtracting numbers and variables of the same kind. After distributing the terms in the expression \( 4F - 2 - 10 \), we focus on combining these like terms.
  • Look for terms that have the same variable. In this case, "like terms" refers to terms with \( F \) such as \( 4F \).
  • Also look at pure numbers or constants like \(-2\) and \(-10\) to combine them separately.
Combining these terms gives us:
\[ 4F - 12 \]
The process of combining like terms simplifies our expression, making it easier to understand and solve. It is a step-by-step process where similar types of terms are added together.
The Simplification Process
Simplification is the process of reducing an expression to its most basic form. In our exercise, after combining like terms within the parentheses, we have to address the initial expression \(-2F +(4F - 12)\).
Here's what happens during simplification:
  • Identify like terms across the entire expression. For instance, identify \(-2F\) and \(4F\) as like terms because they both involve \( F \).
  • Add \(-2F\) and \(4F\) to get \(2F\).
Now we have:
\[ 2F - 12 \]
This is the simplest form of the original expression. The simplification process, therefore, unfolds in stages and each operation brings us closer to a clearer, more straightforward expression. Each step relies on careful arithmetic and understanding of foundational algebraic properties.

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Most popular questions from this chapter

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