The correct answer is B, which demonstrates the Commutative Property of Multiplication. Let's outline why choice B is correct along with an explanation of why the other choices do not demonstrate this property:
Explanation:
The Commutative Property of Multiplication states that changing the order of the factors in a multiplication operation does not change the product; i.e., [tex]\( a \times b = b \times a \)[/tex].
Choice A:
[tex]\[ 6 + (14 + 9) = (6 + 14) + 9 \][/tex]
This is actually demonstrating the Associative Property of Addition, which states that the way in which numbers are grouped when adding does not change the sum.
Choice B:
[tex]\[ \frac{1}{7} \times 9 \times 35 = \frac{1}{7} \times 35 \times 9 \][/tex]
This demonstrates the Commutative Property of Multiplication because it shows that the order in which the multiplication is performed does not affect the result. Here, the factors [tex]\( 9 \)[/tex] and [tex]\( 35 \)[/tex] are reordered, affirming [tex]\( 9 \times 35 = 35 \times 9 \)[/tex].
Choice C:
[tex]\[ \frac{1}{5} \times (15 \times 7) = \left(\frac{1}{5} \times 15\right) \times 7 \][/tex]
This demonstrates the Associative Property of Multiplication, which states that the way in which numbers are grouped when multiplying does not change the product.
Choice D:
[tex]\[ (23 + 9) + 7 = 23 + (9 + 7) \][/tex]
This is demonstrating the Associative Property of Addition, which states that the way in which numbers are grouped when adding does not change the sum.
Choice E:
[tex]\[ 5(x - 7) = 5x - 35 \][/tex]
This demonstrates the Distributive Property of Multiplication over Addition/Subtraction.
Conclusion:
From the explanations above, only choice B shows the Commutative Property of Multiplication by reordering the factors without changing the product. Therefore, the correct answer is choice B.
