Final answer:
The expression x(yz) is equivalent to (xy)(xz) due to the associative property of multiplication. After rearrangement and distribution, both expressions can be shown to result in the same product, confirming the equivalence for any values of x, y, and z.
Step-by-step explanation:
The question asks us to show whether the expression x(yz) is equivalent to (xy)(xz) for all possible inputs of x, y, z. By applying the associative property of multiplication, which states that the way in which factors are grouped does not change the product, we can explore this expression.
Let's consider the expression x(yz). According to the associative property, we can rearrange the parentheses without changing the result: x(yz) = (xy)z. Now, we have the expression in a form that can be further examined.
Next, let's distribute the x across the y and z in the equivalent expression (xy)(xz): xy × xz. By the associative property, we can write this as x(y × xz), or (xy)z, aligning with the rearranged form of our original expression.
Therefore, we can see that x(yz) = (xy)(xz) is true for any values of x, y, and z, which shows that the two expressions are indeed equivalent.