Problem 26 - Beginning and Intermediate Algebra

Question

Decide whether each statement is an example of a commutative, an associative, an identity, \(a n\) inverse, or the distributive property. $$ -4 \cdot(2 \cdot 6)=(-4 \cdot 2) \cdot 6 $$

Step-by-Step Solution

Answer

Associative property of multiplication

Step 1: Understanding the problem

Identify what property is being used in the given mathematical statement \(-4 \cdot (2 \cdot 6) = (-4 \cdot 2) \cdot 6\).

Step 2: Define the property

The associative property of multiplication states that the way in which the factors are grouped does not change the product; i.e., \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).

Step 3: Apply the associative property

Compare the given equation with the associative property: Given: \(-4 \cdot (2 \cdot 6) = (-4 \cdot 2) \cdot 6\) Associative Property: \((a \cdot b) \cdot c = a \cdot (b \cdot c)\) In this problem, the grouping of factors is what changes, while the order of operations and the individual factors remain the same.

Step 4: Conclusion

Since the grouping of factors is what changed in the given statement while maintaining the same order of multiplication, this statement is an example of the associative property of multiplication.

Key Concepts

multiplication properties

Understanding the properties of multiplication is crucial in solving algebraic problems. There are several multiplication properties, but let's focus on the main ones:

• Commutative Property: This states that the order in which two numbers are multiplied does not affect the product. In algebraic terms, this property is written as: \( a \cdot b = b \cdot a \).
• Associative Property: This states that the way numbers are grouped in multiplication does not change the product. For example: \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \).
• Distributive Property: This states that multiplying a number by a sum is the same as multiplying the number by each addend and then adding the products. Written as: \( a \cdot (b + c) = a \cdot b + a \cdot c \).
• Identity Property: This states that any number multiplied by one remains unchanged: \( a \cdot 1 = a \).
• Inverse Property: This states that every number has a reciprocal which, when multiplied together, gives the product of 1: \( a \cdot \frac{1}{a} = 1 \) for \( a eq 0 \).

These properties are fundamental in helping you simplify and solve complex mathematical expressions and equations.

algebraic properties

Algebraic properties are rules that apply to algebraic operations, ensuring consistency and structure in problem-solving. There are a few key properties you should be aware of:

• Addition and Multiplication Properties: These include commutative, associative, and distributive properties, which apply to both addition and multiplication.
• Identity and Inverse Properties: These properties involve the use of 0 in addition (additive identity) and 1 in multiplication (multiplicative identity). Inverse properties involve using negatives in addition and reciprocals in multiplication.
• Zero Product Property: This states that if the product of two numbers is zero, then at least one of the numbers must be zero: \( a \cdot b = 0 \) means either \( a = 0 \) or \( b = 0 \).

By mastering these properties, you can manipulate and solve equations more effectively, making algebra a bit easier to handle.

grouping factors

Grouping factors correctly can help simplify problems and reveal patterns in algebra. The associative property particularly highlights this. Let's dive deeper:
\(\textbf{Associative Property of Multiplication:} \) This property shows that the grouping of numbers in multiplication does not affect the result. Consider the equation: \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \). Here, the product remains the same regardless of how the numbers are grouped, which can be very useful in simplifying expressions and solving equations. For example, in our exercise, we saw: \( -4 \cdot (2 \cdot 6) = (-4 \cdot 2) \cdot 6 \). Both sides yield the same product, confirming that how we group the numbers does not change the result.
Utilizing this property, you can:

• Make calculations easier by re-grouping factors for simplification.
• Break complex problems into smaller, manageable parts.
• Identify patterns and structures to make problem-solving clearer.

It's all about making the math work for you by appropriately using these fundamental properties.