Problem 39 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each expression without parentheses. Then simplify the result. See Example 4 . \(3(6+x)\)

Step-by-Step Solution

Answer

18 + 3x.

Step 1: Identify the Problem

We are given the expression \(3(6+x)\) and need to apply the distributive property to remove the parentheses, then simplify the expression.

Step 2: Apply the Distributive Property

The distributive property states that \(a(b+c) = ab + ac\). Here, \(a=3\), \(b=6\), and \(c=x\). We'll distribute 3 to both 6 and \(x\), resulting in \(3 \cdot 6 + 3 \cdot x\).

Step 3: Perform Multiplication

Calculate \(3 \cdot 6\) and \(3 \cdot x\). We have \(3 \cdot 6 = 18\) and \(3 \cdot x = 3x\).

Step 4: Write the Expanded Expression

Combine the results from Step 3 into a new expression: \(18 + 3x\).

Step 5: Simplify the Expression

The expression \(18 + 3x\) is already simplified, so no further steps are needed.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, symbols, and operations such as addition, subtraction, multiplication, and division. Think of them as sentences that express relationships between quantities. For instance, in the expression \(3(6+x)\), numbers and symbols like 3 and \(x\) are used, along with operations like multiplication and addition. These expressions can be simple or complex.
An expression like \(6 + x\) involves an unknown variable \(x\), which can take different values. By combining numbers (like 6) and variables (like \(x\)), algebraic expressions easily represent a range of possible values. This flexibility is one of the main reasons why algebra is such a powerful tool in mathematics.

Simplification

Simplification in algebra involves reducing an expression to its simplest form. The goal is to make it as concise and understandable as possible. During simplification, we use mathematical properties to combine like terms and eliminate any unnecessary components. This process often makes expressions easier to work with.
For example, in the expression \(18 + 3x\), there are no like terms to combine since 18 is constant and \(3x\) contains the variable \(x\). Thus, simplification is complete, and the expression is already in its simplest form. Simplifying initially complex expressions helps in solving equations, as it reveals the core relationships between the variables involved.

Multiplication Distributive Law

The multiplication distributive law is a fundamental property that helps in multiplying a single term across terms within parentheses. It states: \(a(b+c) = ab + ac\). This law is essential for expanding algebraic expressions so that they can be simplified or evaluated further.
In regard to the problem \(3(6+x)\), this law allows us to distribute 3 through the terms inside the parentheses. First, multiply 3 by 6 to get 18. Then, multiply 3 by \(x\) to get \(3x\). Thus, the expression \(3(6+x)\) simplifies to \(18 + 3x\).

• The distributive law is especially useful in algebra and arithmetic because it simplifies complex expressions into manageable parts.
• It is also foundational in advanced mathematics, as it is central to operations involving polynomials and solving equations.

Understanding this property will make it easier to handle a variety of mathematical problems.