The question asks which expression uses the commutative property of addition and the associative property of multiplication to rewrite the expression $(3 \cdot 2) \cdot 5 + 7$.

To solve this problem, let's analyze each property:

1. Commutative Property of Addition: This property states that changing the order of addition does not change the sum. For example, $a + b = b + a$.

2. Associative Property of Multiplication: This property states that the grouping of factors does not affect the product. For example, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.

Given Expression

The given expression is: $(3 \cdot 2) \cdot 5 + 7$

We need to find an expression that rearranges the numbers using these properties.

Analyzing the Options

1. Option 1: $7 + 3 \cdot (16)$

   • This option changes the addition order ($7 + \ldots$), using the commutative property of addition, but it also involves a multiplication of $3 \cdot 16$, which does not apply the associative property correctly.

2. Option 2: $3 \cdot (5 \cdot 2) + 7$

   • This option uses the associative property of multiplication by changing the grouping to $3 \cdot (5 \cdot 2)$. The addition order remains the same. This matches both properties.

3. Option 3: $7 + 3 \cdot (2 \cdot 5)$

   • This option changes both the addition order and the multiplication grouping, satisfying both the commutative property of addition and associative property of multiplication.

4. Option 4: $(3 \cdot 2) \cdot 5 + 7$

   • This option is the same as the original expression, so it does not demonstrate any use of the commutative or associative properties.

Conclusion

The correct option is Option 3: $7 + 3 \cdot (2 \cdot 5)$, as it correctly uses both the commutative property of addition (changing the order of $7 + \ldots$) and the associative property of multiplication (changing the grouping of $(2 \cdot 5)$).

Would you like further details or have any questions?

Related Questions:

1. What is the difference between commutative and associative properties?
2. Can these properties be applied to subtraction and division?
3. How do distributive properties work with addition and multiplication?
4. What are some practical applications of these mathematical properties?
5. Can these properties be extended to more than three terms?

Tip:

Understanding these fundamental properties of arithmetic can greatly simplify the manipulation of algebraic expressions and solving equations!