Problem 21 - Beginning and Intermediate Algebra

Question

Use a commutative or an associative property to complete each statement. State which property is used. \(7 \cdot(2 \cdot 5)=(\) ____ \(\cdot 2) \cdot 5\)

Step-by-Step Solution

Answer

The property used is Associative Property. The completed statement is \((7 \times 2) \times 5\).

Step 1: Identify the Problem Structure

The problem presents a multiplication expression involving three numbers: 7, 2, and 5. It needs to be restructured using either the commutative property or the associative property.

Step 2: Recognize the Required Property

The expression can be rearranged using the associative property of multiplication. The associative property states that the way in which factors are grouped in a multiplication expression does not change the product.

Step 3: Apply the Associative Property

According to the associative property, \(a \times (b \times c) = (a \times b) \times c\). Here, we can group 7 with 2, then multiply the result by 5.

Step 4: Rewrite the Expression

Using the associative property, the original expression \(7 \times (2 \times 5)\) becomes \((7 \times 2) \times 5\).

Step 5: Complete the Statement

Therefore, the completed statement is \(7 \times (2 \times 5) = (7 \times 2) \times 5\).

Key Concepts

Commutative Property

The commutative property is an essential concept in arithmetic, especially for operations such as addition and multiplication. This property states that changing the order of the numbers involved does not change the result. In terms of multiplication, it means that if you switch the places of the factors, the product remains the same. For example, if you have the expression 3 × 4, using the commutative property, you can rearrange it as 4 × 3. Both expressions will equal 12. So, for any numbers a and b, the commutative property of multiplication can be written as:

• a × b = b × a.

Understanding the commutative property helps simplify and solve more complex arithmetic problems. It emphasizes that the order of factors does not affect the outcome, which can be handy in break-down computations and make mental math easier.

Multiplication

Multiplication is one of the fundamental arithmetic operations. It involves combining equal groups to find out how many items there are in total. For example, if you have 3 groups of 4 apples, you can quickly find out there are 12 apples by multiplying 3 by 4. Multiplication can be used to scale numbers up, making it quicker to add large groups together. The properties of multiplication, such as the commutative and associative properties, play crucial roles in solving multiplication problems efficiently. Understanding these properties can help lessen the complexity and enhance the accuracy of your calculations. Always keep in mind the symbols and steps involved in multiplication for effective learning.

Expression Rearrangement

Rearranging expressions is an essential skill in algebra and arithmetic. It involves changing the arrangement of numbers or terms to a more simplified or convenient form without changing the value of the expression. For instance, given a multiplication problem, you might use properties like the associative or commutative properties to rearrange the factors for easier computation.

• The associative property allows you to change the grouping of the factors. For example, (7 × 2) × 3 can be written as 7 × (2 × 3).
• The commutative property permits swapping the order of the factors, like turning 2 × 7 into 7 × 2.

These rearrangements help simplify your calculations and solve the problem with less effort. In the given example, using the associative property, the expression 7 × (2 × 5) was rearranged as (7 × 2) × 5, making it easier to understand and solve. Practice these rearrangement techniques to become proficient in handling more complex algebraic expressions.