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Chapter 2: Problem 106

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

Short Answer

Expert verified
The simplified expression is \(4a + 4b - 4ab\).

Step by step solution

01

Identify the Expression

The expression given is \(4(a+b)+4a(-b)\).
02

Distribute the Coefficients

Apply the distributive property to both terms: \(4(a+b)\) becomes \(4a + 4b\) and \(4a(-b)\) becomes \(4a \times -b = -4ab\). Thus, the expression becomes \(4a + 4b - 4ab\).
03

Combine Like Terms

Identify like terms in the expression \(4a + 4b - 4ab\). There are no like terms to combine, as each term is distinct with different variables and exponents.
04

Simplify the Expression

Since there are no like terms to combine further, the simplified expression remains \(4a + 4b - 4ab\).

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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 foundation in algebra. It lets you multiply a single term by each term within a parenthesis. It simplifies expressions and makes them easier to work with.

For example, in the expression given: \(4(a + b) + 4a(-b)\), you need to distribute the number outside the parenthesis to each term inside.

Let's see it step-by-step:
  • Distribute 4 to \(a + b\): \(4a + 4b\).
  • Distribute 4a to \(-b\): \(-4ab\).
Now, the original expression becomes \(4a + 4b - 4ab\). This process helps break down complex problems into simpler parts.
Like Terms
Like terms are key in combining terms. They have the same variables and exponents. Identifying them is the first step in simplifying an expression.

For instance, in the expression \(4a + 4b - 4ab\), look for terms with the same variable factors.
  • 4a and 4b are not like terms because they have different variables (a and b).
  • -4ab also stands alone as it has both variables (a and b) multiplied together.
Knowing what qualifies as like terms ensures you simplify expressions correctly. Look closely at both the variables and their exponents.
Combining Terms
Once you identify like terms, the next step is to combine them. This makes your expression shorter and simpler.

To combine:
  • Add or subtract their coefficients (numbers in front of variables).
  • Keep the variable part unchanged.
However, in the expression \(4a + 4b - 4ab\), no like terms exist.

Because of different variables and combinations, they remain as they are. No further combination happens here. This step only works when there are like terms to add or subtract.
Algebraic Simplification
Algebraic simplification pulls all these core actions together. It aims to create the simplest form of an expression, making it easier to solve or understand.

Combine like terms, use distributive property, and maintain order of operations. Even if no combination is possible, as in \(4a + 4b - 4ab\), recognizing every simplification step matters.

Simplification helps in:
  • Solving equations faster.
  • Clearer understanding of mathematical relationships.
  • Avoiding errors in longer calculations.
Practice these steps consistently to build your algebra skills. You'll notice complex problems becoming easier over time.

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Most popular questions from this chapter

Separate each list into groups of like terms, and name the coefficient and literal part of each term. $$4, \quad 4 u, \quad 4 u^{2}, \quad 5, \quad 5 u$$

How can the distributive property be used to make it easier to compute the following expressions? (a) \(45(116)-45(16)\) (b) \(60(278)-60(28)\)

Ruth walks at the rate of 100 meters per minute and jogs at the rate of 220 meters per minute. She walks for 25 minutes and then jogs for 35 minutes. (a) How fast does she walk? (b) For how long does she walk? (c) How far does she walk? (d) How fast does she jog? (e) For how long does she jog? (f) How far does she jog? (g) How much distance has Ruth covered all together?

Evaluate the given expression for \(x=2, y=-3,\) and \(z=-4.\) $$x y z^{2}$$

Simplify by combining like terms whenever possible. $$4 a-2 b-5 c$$
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