Problem Analysis:
The task involves identifying the type of property being used in a given mathematical statement. The properties mentioned are:
1.
Commutative Property: This property states that the order of operands does not affect the result. For example:
- Addition: \( a + b = b + a \)
- Multiplication: \( a \times b = b \times a \)
2.
Associative Property: This property states that the grouping of operands does not affect the result. For example:
- Addition: \( (a + b) + c = a + (b + c) \)
- Multiplication: \( (a \times b) \times c = a \times (b \times c) \)
3.
Distributive Property: This property involves distributing one operation over another, typically multiplication over addition or subtraction. For example:
- \( a \times (b + c) = a \times b + a \times c \)
- \( a \times (b - c) = a \times b - a \times c \)
The problem provides an example and asks to identify which property is being used.
---
Given Example:
The example provided is:
\[
(8 \times 2) + (8 \times 7) = 8 \times (2 + 7)
\]
#### Step-by-Step Solution:
1.
Identify the Structure:
- On the left-hand side: \( (8 \times 2) + (8 \times 7) \)
- On the right-hand side: \( 8 \times (2 + 7) \)
2.
Analyze the Operation:
- The left-hand side involves two separate multiplications (\( 8 \times 2 \) and \( 8 \times 7 \)) added together.
- The right-hand side involves a single multiplication where \( 8 \) is multiplied by the sum of \( 2 \) and \( 7 \).
3.
Match with Properties:
-
Commutative Property: This property deals with changing the order of operands, but here the order is not being changed; instead, the structure of the expression is being altered.
-
Associative Property: This property deals with changing the grouping of operands, but here the grouping is not being changed; instead, the distributive nature of multiplication over addition is being applied.
-
Distributive Property: This property matches perfectly because it describes how multiplication distributes over addition. Specifically:
\[
a \times (b + c) = (a \times b) + (a \times c)
\]
Here, \( a = 8 \), \( b = 2 \), and \( c = 7 \). The given equation is a direct application of the distributive property.
4.
Conclusion:
- The property being used in the given example is the
Distributive Property.
---
Final Answer:
\[
\boxed{\text{Distributive Property}}
\]
Parent Tip: Review the logic above to help your child master the concept of commutative associative and distributive properties worksheet.
