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Chapter 2: Problem 104

Simplify by combining like terms whenever possible. $$4(a+b)+4(a-b)$$

Short Answer

Expert verified
8a

Step by step solution

01

Distribute the 4 across the terms inside the parentheses

Apply the distributive property to each term inside the parentheses. This means multiplying 4 by each term inside the parentheses: 4(a+b) = 4a + 4b and 4(a-b) = 4a - 4b.
02

Write down the expanded terms

Combine the results from Step 1: 4a + 4b + 4a - 4b.
03

Combine like terms

Identify and add together the like terms. The like terms here are 4a and 4a, and 4b and -4b. 4a + 4a + 4b - 4b = 8a.

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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 helps you simplify expressions by getting rid of parentheses. You apply this property by multiplying a single term by each term inside the parentheses. For our example, the distributive property is used twice:

  • For the expression \(4(a+b)\), we multiply 4 by both \(a\) and \(b\), which gives us \(4a + 4b\).
  • Similarly, for \(4(a-b)\), we multiply 4 by both \(a\) and \(b\), but since \(b\) has a negative sign in front, it becomes \(4a - 4b\).
The distributive property helps break down complex expressions into simpler terms so you can see what you are working with more clearly.
Combining Like Terms
Combining like terms is another fundamental skill in algebra. Like terms are terms that have the same variable raised to the same power. In our example, look at \(4a\), \(4b\), \(4a\), and \(-4b\).

  • First, pair up the terms with the same variables: \(4a\) and \(4a\); \(4b\) and \(-4b\).
  • Next, add or subtract these pairs: \(4a\) plus \(4a\) equals \(8a\) and \(4b\) minus \(4b\) equals 0.
When you combine like terms, you simplify the expression further, making it easier to understand and solve.
Algebraic Simplification
Simplifying algebraic expressions involves using both the distributive property and combining like terms to reduce an expression to its simplest form. Let's revisit our example:

  • After applying the distributive property, you get: \(4a + 4b + 4a - 4b\).
  • Then, by combining like terms, the expression simplifies to \(8a\).
Why do we simplify? Simplification makes equations easier to solve and understand. It highlights the fundamental quantities you're working with, reducing the chance of errors in subsequent steps.

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Most popular questions from this chapter

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Simplify by combining like terms whenever possible. $$3(x+y)+4(x-y)$$
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