To determine which of the following demonstrates the associative property, we need to review what the associative property is. The associative property refers to the grouping of numbers in addition or multiplication. According to this property:
For addition: [tex]\( a + (b + c) = (a + b) + c \)[/tex]
For multiplication: [tex]\( a \times (b \times c) = (a \times b) \times c \)[/tex]
Now, let's evaluate each option:
- Option A: [tex]\( x(y x) = x y z \)[/tex]
- This expression is malformed and does not make sense in the context of the associative property. There is no clear way of grouping or correctly placing multiplication.
- Option B: [tex]\( x(y z) = (x y) z \)[/tex]
- This demonstrates the associative property of multiplication correctly. The grouping of [tex]\( y \)[/tex] and [tex]\( z \)[/tex], and [tex]\( x \)[/tex] and [tex]\( y \)[/tex], does not change the result.
- Option C: [tex]\( x y z = z y x \)[/tex]
- This demonstrates the commutative property, where the order of multiplication does not matter. It is not related to the associative property, which concerns grouping.
- Option D: [tex]\( x y \times 0 = 0 \)[/tex]
- This describes the multiplication property concerning zero, stating that any number multiplied by zero equals zero. It does not describe the associative property.
Based on this analysis, the correct option that demonstrates the associative property is:
B. [tex]\( x(y z) = (x y) z \)[/tex]
