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Chapter 0: Problem 12

Simplify each expression. $$ 3(a+b) $$

Short Answer

Expert verified
The simplified expression is \( 3a + 3b \).

Step by step solution

01

Identify the Expression

The given expression is \( 3(a+b) \).
02

Apply the Distributive Property

Use the distributive property of multiplication over addition. The distributive property states that \( a(b + c) = ab + ac \). Here, apply it to \( 3(a+b) \).
03

Distribute and Simplify

Distribute the 3 to both \( a \) and \( b \). Therefore, \( 3(a+b) = 3a + 3b \).
04

Conclusion

The simplified expression is \( 3a + 3b \).

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is a fundamental skill in algebra. This process involves transforming a given mathematical expression into its simplest form by following a set of rules and operations. One important rule used in simplification is the distributive property.
For instance, consider the given expression: \( 3(a+b) \). To simplify it, we apply the distributive property which allows us to multiply each term inside the parentheses by the factor outside. This is shown in the steps:
  • Step 1: Identify the Expression The given expression is \( 3(a+b) \).
  • Step 2: Apply the Distributive Property The distributive property states that \( a(b + c) = ab + ac \). In this case, \( a = 3 \), \( b = a \), and \( c = b \).
  • Step 3: Distribute and Simplify Multiply 3 by both \( a \) and \( b \), resulting in \( 3a + 3b \).
  • Step 4: Conclusion The simplified expression is \( 3a + 3b \).
Simplifying algebraic expressions helps in making complex operations more manageable and is a cornerstone for solving equations.
Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and arithmetic operations. Variables represent unknown values and can be any letter like \(a\) or \(b\). In the expression \(3(a + b)\), \(a\) and \(b\) are variables, while \(3\) is a coefficient.
To work with algebraic expressions, you often need to combine like terms or use properties such as distributive, associative, and commutative properties. For example:
  • Distributive Property: Allows you to expand expressions like \(3(a + b)\) into \(3a + 3b\).
  • Associative Property: States that how you group numbers does not change their sum or product, i.e., \((a + b) + c = a + (b + c)\).
  • Commutative Property: States that the order in which you add or multiply numbers does not matter, i.e., \(a + b = b + a\) and \(ab = ba\).
Understanding these properties helps in manipulating and solving algebraic expressions efficiently.
Basic Algebra Concepts
Basic algebra concepts form the foundation of more advanced mathematical studies. Here are a few key concepts:
  • Variables: Symbols used to represent unknown values. In the expression \(3(a + b)\), \(a\) and \(b\) are variables.
  • Coefficients: Numbers multiplying the variables. In \(3a\), \(3\) is the coefficient.
  • Constants: Fixed values that do not change, like numbers without variables attached.
  • Expressions: Combinations of variables, coefficients, and constants, like \(3(a + b)\).
To master algebra, practice simplifying and manipulating these expressions, using fundamental principles like the distributive property. Each manipulation helps prepare you for solving equations and understanding more complex math topics.
Working with basic algebra concepts boosts problem-solving skills and confidence in handling mathematical challenges.

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