Problem 72 - Prealgebra

Question

Apply the distributive property to each expression. $$5(3+7)$$

Step-by-Step Solution

Answer

The expression simplifies to 50.

Step 1: Identify the Terms Inside the Parentheses

In the expression \(5(3+7)\), recognize that \(3\) and \(7\) are the terms inside the parentheses that need to be distributed.

Step 2: Apply the Distributive Property

The distributive property states that \(a(b+c) = ab + ac\). Let's apply it here: \(5(3+7) = 5 \times 3 + 5 \times 7\).

Step 3: Multiply Each Term

Calculate the multiplication for each part: \(5 \times 3 = 15\) and \(5 \times 7 = 35\).

Step 4: Add the Results

Add the results from Step 3: \(15 + 35 = 50\).

Key Concepts

Mathematical Expressions

Mathematical expressions are combinations of numbers, operations, and sometimes variables, that represent a value or set of values. In this context, our expression is \(5(3+7)\). Understanding how to manipulate expressions using properties like the distributive property is essential for simplifying them and solving equations.

• Basic Components: A mathematical expression may include numbers (also known as constants), variables (unknown values usually represented by letters), and operations (such as addition, subtraction, multiplication, and division).
• Understanding Parentheses: Parentheses are used in expressions to indicate which operations should be performed first according to the rules of operations, known as the order of operations.
• Distributive Property In Action: In the expression \(5(3+7)\), parentheses group \(3+7\), suggesting that this operation should be evaluated first. However, the distributive property allows us to multiply each individual term within the parentheses by the number outside, which in this case is 5.

Recognizing and understanding the structure of mathematical expressions is a fundamental skill in mathematics that forms the basis for more advanced topics.

Prealgebra

Prealgebra is the branch of mathematics that prepares students for algebra by reviewing arithmetic along with introducing fundamental algebra concepts. It is during this stage that the distributive property becomes particularly relevant.

• Foundation of Algebra: Prealgebra lays the groundwork for algebra by introducing students to variables and expressions. Learning the distributive property helps in understanding how to handle expressions and simplifying them.
• Arithmetic to Algebra Transition: Key concepts like the distributive property bridge the gap between the familiar world of arithmetic and the more abstract realm of algebra. This transition is crucial as it sets the stage for solving equations and inequalities in algebra.
• Developing Problem-Solving Skills: By applying the distributive property, students practice breaking down problems into smaller, manageable parts, which improves their problem-solving abilities.

Prealgebra acts as a guide for students to become confident in approaching algebra, which involves more complex operations and equations.

Step-by-Step Solutions

Step-by-step solutions provide a systematic approach to solving mathematical problems, ensuring that students follow a structured method and understand each stage involved in solving an exercise such as using the distributive property.

• Clarity and Structure: Breaking down a problem into steps allows students to focus on one part of the problem at a time. This helps in understanding the sequence of operations necessary to reach a solution.
• Building Confidence: When students consistently work through problems step-by-step, they gain confidence in their abilities to tackle similar problems independently.
• Identifying Mistakes: A clear, detailed solution enables students to see where they may have made errors, providing an opportunity for learning and improvement.

Approaching problems in a step-by-step manner is especially useful in learning new mathematical concepts and applying them effectively, as demonstrated in the exercise above with the distribution across terms.