Problem 17 - Beginning and Intermediate Algebra

Question

Use a commutative or an associative property to complete each statement. State which property is used. \(-8 \cdot 3=\) ____ \(\cdot(-8)\)

Step-by-Step Solution

Answer

3 \times (-8), Commutative property of multiplication

Step 1: Identify the equation

Examine the equation \(-8 \times 3 =\) ____ \( \times (-8)\) this can be rewritten using a property.

Step 2: Recognize the commutative property of multiplication

Recall that the commutative property of multiplication states that for any numbers \(a\) and \(b\), \(a \times b = b \times a\).

Step 3: Apply the commutative property

Swap the order of the numbers using the commutative property to get: \(-8 \times 3 = 3 \times (-8)\)

Key Concepts

Commutative Property

The commutative property of multiplication is a fundamental concept in mathematics.
It states that the order in which you multiply two numbers does not affect the product.
For any two numbers, say \(a\) and \(b\), the property is expressed as: \(a \times b = b \times a\).
This means that if you multiply \(-8\) and \(3\), the result is the same whether you do \(-8 \times 3\) or \(3 \times (-8)\).
This property is very useful, especially when simplifying complex equations or checking your work.
So in our exercise, you can see how applying the commutative property transforms \(-8 \times 3\) into \(3 \times (-8)\).

Associative Property

The associative property of multiplication is another key principle to understand.
This property focuses on the grouping of numbers and states that the way numbers are grouped in multiplication does not change their product.
For any three numbers, say \(a\), \(b\), and \(c\), the associative property is expressed as: \((a \times b) \times c = a \times (b \times c)\).
This means you can change the grouping of the numbers being multiplied without changing the result.
While this property is not directly used in the example problem, it is equally important as it allows flexibility in rearranging factors to simplify calculations.
When dealing with multiple factors, remembering the associative property can make your work a lot easier.

Multiplication

Multiplication is one of the most basic arithmetic operations.
It is essentially repeated addition; multiplying two numbers involves adding one of the numbers to itself as many times as the value of the other number.
For example, \(3 \times 4\) means adding \(3\) four times: \(3 + 3 + 3 + 3 = 12\).
In terms of properties, multiplication has two important properties: commutative and associative.
These properties help us manipulate and simplify expressions, making the operations faster and reducing the chances of errors.
Understanding these properties well can significantly enhance your mathematical skills and problem-solving abilities.