Problem 33 - Prealgebra

Question

Apply the distributive property to each expression and then simplify. $$2(3 x+4)+8$$

Step-by-Step Solution

Answer

The simplified expression is \(6x + 16\).

Step 1: Apply the Distributive Property

First, apply the distributive property to the term \(2(3x + 4)\). This means you will multiply each term inside the parentheses by 2. Thus, \(2 \times 3x = 6x\) and \(2 \times 4 = 8\). Rewrite the expression as \(6x + 8 + 8\).

Step 2: Simplify the Expression

Combine the like terms in the expression \(6x + 8 + 8\). The terms \(8 + 8\) are constant terms and can be added together to get 16. Therefore, the simplified expression is \(6x + 16\).

Key Concepts

Simplifying Expressions

Simplifying expressions is the process of making a mathematical expression more manageable and easier to work with. It involves performing operations and re-writing expressions in their simplest form. In the given problem, we start by applying the distributive property which allows us to simplify the expression step-by-step.

After using the distributive property on the expression \(2(3x + 4)\), we end up with \(6x + 8 + 8\). The next step is to simplify the expression by performing all possible operations.

• First, look for any like terms in the expression.
• Add, subtract, multiply, or divide to transform the expression into its simplest form.

Simplifying expressions is fundamental for solving equations and allows for a clearer view of what the expression represents. It reduces potential errors in calculations and makes further algebraic work easier.

Combining Like Terms

Combining like terms is an essential step in simplifying expressions. Like terms in algebra are terms that have the same variable raised to the same power. In our exercise, like terms must be identified and combined to simplify the expression further.

In the resulting expression \(6x + 8 + 8\), the numbers 8 and 8 are like terms because they are both constants. By adding them together, we simplify the expression to \(6x + 16\).

• Always look for terms with the same variables and exponents.
• Combining these terms reduces the number of terms in the expression, making it simpler.
• In coefficients, simply perform the required operation (such as addition or subtraction).

This process is particularly helpful when dealing with larger, more complex expressions, as it reduces the clutter and makes subsequent calculations less error-prone.

Prealgebra Concepts

Understanding prealgebra concepts, such as the distributive property and combining like terms, lays the groundwork for more advanced algebraic work. These concepts are pivotal when dealing with arithmetic operations involving variables.

The distributive property is of significant importance in algebra. It states that for any numbers or variables \(a, b\) and \(c\), the expression \(a(b + c)\) equals \(ab + ac\). This property allows us to distribute a multiplier across terms inside parentheses and is an indispensable tool for expanding and simplifying expressions.

• These foundational concepts help in solving equations efficiently.
• They simplify the expressions, paving the way for further operations.
• A strong understanding of these concepts aids in grasping more complex algebra topics.

Therefore, mastering prealgebra concepts is crucial for anyone embarking on the study of algebra, as they form the basis for high school mathematics and beyond.