Problem 18 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent expression. Then evaluate it. $$(4+3) 3$$

Step-by-Step Solution

Answer

Using distributive property, the equivalent expression is \(4\times3 + 3\times3\); its value is 21.

Step 1: Identify the Distributive Property

The distributive property states that for any three numbers, \( a \), \( b \), and \( c \), the expression \( a(b + c) \) is equivalent to \( ab + ac \). In this expression, you need to distribute the 3 across the sum inside the parentheses.

Step 2: Apply the Distributive Property

Use the distributive property on \( (4 + 3) \times 3 \) by multiplying each term inside the parentheses by 3. This gives us: \( 4 \times 3 + 3 \times 3 \).

Step 3: Simplify Each Term

Calculate each individual product: \( 4 \times 3 = 12 \) and \( 3 \times 3 = 9 \).

Step 4: Add the Products

Add the results from the previous step: \( 12 + 9 = 21 \). Thus, the value of the expression is 21.

Key Concepts

Equivalent Expressions

The concept of equivalent expressions is central to understanding algebra. When we talk about expressions being "equivalent," we mean that even though the expressions might look different, they have the same value. In mathematics, you can transform an expression using various properties and rules without changing the expression's value. One important property for creating equivalent expressions is the Distributive Property.

For example, in the expression \((4 + 3) \times 3\), using the Distributive Property allows us to break it down to \(4 \times 3 + 3 \times 3\). Even though these expressions look different, they carry the same outcome when evaluated. This property helps in simplifying expressions, making complex calculations more manageable.

Arithmetic

Arithmetic forms the foundation of math learning, focusing on basic operations like addition, subtraction, multiplication, and division. In the context of our expression \((4 + 3) \times 3\), arithmetic skills are crucial for performing the operations accurately.

Here's how it works for this problem:

• We start by distributing the multiplication over addition, using the Distributive Property. This helps in breaking down the expression into simpler parts: \(4 \times 3 + 3 \times 3\).
• Next, we compute each multiplication separately. First, \(4 \times 3 = 12\). Then, \(3 \times 3 = 9\).
• Finally, we add these products together: \(12 + 9\) which results in \(21\).

These steps highlight how arithmetic rules and processes are used to simplify and solve expressions effectively.

Mathematical Properties

Mathematical properties are rules that apply to numbers and assist in the manipulation and simplification of expressions. Some of these properties include the Commutative, Associative, Identity, and Distributive Properties. Here, we're focusing on the Distributive Property, which is key in transforming and simplifying expressions.

The Distributive Property tells us that you can "distribute" a multiplication over an addition within a parenthesis. This can be written as: \(a(b + c) = ab + ac\). In our original exercise, we have:\((4 + 3) \times 3\). By distributing, we rewrite it as \(4 \times 3 + 3 \times 3\), which simplifies our calculations.

Effectively using these mathematical properties helps make complex arithmetic simpler, providing a straightforward method to evaluate more challenging expressions. Understanding these properties will enhance your skills, making it easier to manipulate and solve a broad array of mathematical problems.