Problem 10 - Intermediate Algebra

Question

State the property that justifies each of the statements. For example, \(3+(-4)=(-4)+3\) because of the commutative property of addition. $$ [(-7)(4)](-25)=(-7)[4(-25)] $$

Step-by-Step Solution

Answer

Associative property of multiplication.

Step 1: Identify the Statement Structure

The given statement is \([(-7)(4)](-25)=(-7)[4(-25)]\). It involves rearranging the parentheses in a multiplication expression without altering the numbers involved.

Step 2: Determine Relevant Mathematical Property

The statement shows that the grouping of numbers in multiplication changes, recognizing this as the associative property of multiplication.

Step 3: Review the Associative Property of Multiplication

The associative property states that changing the grouping of numbers does not change their product. In formula form: \( (a \times b) \times c = a \times (b \times c) \).

Step 4: Conclude the Justification

Thus, the expression \([(-7)(4)](-25)=(-7)[4(-25)]\) is justified by the associative property of multiplication, as it merely changes the grouping of factors.

Key Concepts

Commutative Property

The commutative property in mathematics tells us a simple yet fundamental rule about numbers in arithmetic operations like addition and multiplication. Simply put, it means that the order in which you multiply or add numbers does not change the result.
Let's look in detail at how this works for multiplication, which is our focus here:

• For two numbers, say "a" and "b", the property showcases that: \( a \times b = b \times a \).
• Regardless of which number you put first, the product remains the same. This helps add versatility when solving problems because it allows numbers to be rearranged in a way that's more convenient or simpler for us to work with.

Understanding the commutative property helps greatly when solving complex equations as you can reorganize terms freely without any consequence to the final answer.

Grouping of Numbers

When discussing the grouping of numbers in mathematics, we often refer to various methods of arranging numbers in calculations, particularly in operations like addition and multiplication. It is essential to understand how this grouping operates, especially when dealing with long expressions.
The associative property, which highlights the concept of grouping, allows us to change the way numbers are grouped without affecting the final outcome.

• In our context of multiplication, this can be expressed as: \( (a \times b) \times c = a \times (b \times c) \).
• This means whether you multiply the first pair of numbers first, or regroup and start with another pair, you'll end up with the same product.

The ability to re-group numbers freely can greatly simplify working through complex equations by focusing on easier calculations or recognizing patterns that simplify the problem.

Multiplication Expression

A multiplication expression is simply a mathematical phrase that involves numbers and multiplication operations. These expressions can be simple or complex, involving whole numbers, decimals, or fractions, and sometimes require the use of properties to solve or simplify.
These expressions make use of arithmetic properties such as the associative and commutative properties.

• The associative property allows us to change how we group numbers, making it easier to handle computations in steps.
• The commutative property assures us that the order of numbers won't affect the final result, making rearrangement possible for convenience or by strategic necessity.

Understanding these properties in a multiplication expression can be incredibly useful. They provide the tools needed to rearrange and solve expressions in ways that are often more straightforward or practical than trying to solve them linearly. This adaptability is invaluable, especially in more advanced mathematical problem-solving scenarios.