Problem 71 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The result is 50 after applying the distributive property.

Step 1: Recognize the Distributive Property

The distributive property states that for any numbers \( a \), \( b \), and \( c \), the expression \( a(b + c) \) is equivalent to \( ab + ac \). In this exercise, we need to apply this property to the expression \( 5(3 + 7) \).

Step 2: Apply the Distributive Property

According to the distributive property, distribute the \( 5 \) to both terms inside the parentheses. This means multiplying \( 5 \) by \( 3 \), and \( 5 \) by \( 7 \), giving us: \( 5 \times 3 + 5 \times 7 \).

Step 3: Calculate the Products

Calculate each product from the distribution: \( 5 \times 3 = 15 \) and \( 5 \times 7 = 35 \).

Step 4: Add the Results

Add the products obtained from multiplying: \( 15 + 35 = 50 \).

Key Concepts

Prealgebra

Prealgebra is a fundamental stepping stone in mathematics that prepares students for more advanced topics like algebra and beyond. It deals primarily with basic arithmetic operations and introduces students to the idea of variables and simple equations. In the context of our exercise, prealgebra provides the foundation needed to understand and apply the distributive property.

• It helps students build confidence in solving mathematical problems.
• It aids in developing logical thinking and problem-solving skills.

By mastering prealgebra concepts, students set themselves up for success as they progress to tackling more complex algebraic expressions.

Multiplication

Multiplication is one of the four basic arithmetic operations. In the context of the distributive property, it's essential to understand how to multiply numbers correctly. When we say apply the distributive property, we mean distributing one term to each term inside the parentheses by multiplying.

• If you have a term outside the parentheses, like the number 5 in our expression, you need to multiply it by each term inside the parentheses.
• This means calculating products such as multiplying 5 by 3 and then by 7 in our example.

This operation is crucial as it simplifies the expression, making it easier to solve without expanding the entire expression at once. Moreover, it shows how multiplication can be distributed over addition, reinforcing the power and versatility of multiplication in arithmetic.

Addition

Addition is another core arithmetic operation and a fundamental part of the prealgebraic process. Once we've multiplied the individual terms using the distributive property, what comes next is adding these results together. Adding the products obtained, such as 15 and 35, culminates in the final step of simplifying the expression. This step is crucial to obtaining the correct final value when applying the distributive property.

• For our problem, addition means simply calculating 15 plus 35.
• This final addition gives us the resultant of our expression, which in our case is 50.

Through practice, students not only gain proficiency in performing these calculations but also understand how different arithmetic operations interplay in expressions.

Simplify Expressions

Simplifying expressions involves making an expression easier to read or solve, reducing it to its simplest form. Using the distributive property is one way to achieve this in prealgebra. For the expression involving the distributive property, simplification is the process of:

• Distributing the multiplication over the addition inside the parentheses.
• Calculating the necessary products.
• Combining these through addition to get a final, simpler value.

This approach helps students break down seemingly complex expressions into manageable parts, leading to clearer solutions and a deeper understanding of how different operations connect within mathematics. Over time, mastering the skill of simplifying expressions allows students to approach mathematical problems with greater ease and confidence.