To figure out which mathematical property is exemplified by the expression [tex]\((3)[5(7)] = (3)[7(5)]\)[/tex], let's analyze what each property listed is and how it relates to the given expression.
1. Associative Property:
- Addition: [tex]\((a + b) + c = a + (b + c)\)[/tex]
- Multiplication: [tex]\((a \cdot b) \cdot c = a \cdot (b \cdot c)\)[/tex]
- The associative property deals with grouping of numbers. Changing the grouping of numbers does not change the result.
2. Commutative Property:
- Addition: [tex]\(a + b = b + a\)[/tex]
- Multiplication: [tex]\(a \cdot b = b \cdot a\)[/tex]
- The commutative property deals with the order of numbers. Changing the order of the numbers does not change the result.
3. Distributive Property:
- [tex]\(a(b + c) = ab + ac\)[/tex]
- The distributive property shows how multiplication can be distributed over addition or subtraction.
4. Identity Property:
- Addition: [tex]\(a + 0 = a\)[/tex]
- Multiplication: [tex]\(a \cdot 1 = a\)[/tex]
- The identity property applies when adding zero to a number or multiplying a number by one leaves the number unchanged.
Given the expression [tex]\((3)[5(7)] = (3)[7(5)]\)[/tex]:
- We see two sets of numbers inside nested parentheses: [tex]\((5 \cdot 7)\)[/tex] and [tex]\((7 \cdot 5)\)[/tex].
- The expression suggests that changing the order of [tex]\(5\)[/tex] and [tex]\(7\)[/tex] within the operations does not affect the outcome.
This corresponds to the commutative property, which states that the order of numbers can be changed without affecting the result.
Thus, the property shown in the given expression is the commutative property.
