Problem 32 - Algebra A Combined Function

Question

Use the distributive property to write each expression without parentheses. Then simplify the result, if possible. See Examples 7 through 12. $$ 7(a+b) $$

Step-by-Step Solution

Answer

The expression \(7(a+b)\) simplifies to \(7a + 7b\).

Step 1: Apply the Distributive Property

To apply the distributive property, multiply each term inside the parentheses by the term outside the parentheses. For the expression \(7(a+b)\), distribute \(7\) to both \(a\) and \(b\). Thus, \(7(a+b) = 7 \cdot a + 7 \cdot b\).

Step 2: Simplify the Expression

Now that the distributive property is applied, simplify the expression by performing the multiplication indicated. \(7 \cdot a = 7a\) and \(7 \cdot b = 7b\). So, the expression \(7(a+b)\) becomes \(7a + 7b\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that helps in transforming complex problems into more manageable ones. Essentially, it involves performing operations like addition, subtraction, multiplication, or division to combine like terms for a clearer and more concise form. For example, if you have an expression like \(7(a+b)\), simplifying it using the distributive property will result in \(7a + 7b\). This step is crucial for making future calculations easier and more intuitive.

When simplifying, focus on the following:

• Combine like terms: Only terms with the same variables and exponents can be combined.
• Perform basic operations: Follow the order of operations—remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Breaking down an equation step-by-step helps avoid errors and makes algebraic expressions easier to handle in more complex problems.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. They are the building blocks of algebra. The expression \(7(a+b)\) is an example where 7, \(a\), and \(b\) are the basic components.

Key features to note about algebraic expressions include:

• Variables: Symbols like \(a\) and \(b\) which represent unknowns.
• Coefficients: Numerical factors like 7 in \(7(a+b)\) that multiply the variable(s).
• Operators: Signs such as + or * which indicate the operations performed between numbers and variables.

Understanding the structure of algebraic expressions is pivotal. It allows students to effectively manipulate and solve equations by applying algebraic rules and properties.

Multiplication of Terms

Multiplying terms, especially within algebraic expressions, can be straightforward once the laws of arithmetic and algebra are understood. Multiplication is the process used to increase a number repeatedly by another number. In the expression \(7(a+b)\), this involves using the distributive property to multiply 7 by each of the terms inside the parentheses, resulting in \(7a + 7b\).

Here's how multiplication of terms works effectively:

• Identity Law: Any term multiplied by 1 equals the term itself, for example, \(1 \cdot a = a\).
• Zero Product Property: Any term multiplied by 0 equals 0, for example, \(0 \cdot a = 0\).
• Distributive Property: This vital property states that \(a(b+c) = ab + ac\).

By following these principles, multiplying terms in algebra becomes a mechanical and predictable process, leading to an accurate simplification of any given expression.