When studying properties of operations, especially in the context of complex numbers, it's important to understand the fundamental properties that apply. Here, we are given that [tex]\(x = a + bi\)[/tex] and [tex]\(y = c + di\)[/tex] and asked to consider [tex]\(x \cdot y = y \cdot x\)[/tex].
We need to identify the property that describes the equality of [tex]\(x \cdot y\)[/tex] and [tex]\(y \cdot x\)[/tex].
Consider the following properties:
1. The commutative property states that the order of the operands does not affect the result. For example, [tex]\(a \cdot b = b \cdot a\)[/tex].
2. The identity property refers to the existence of an identity element that does not change the value of the other operand when used in an operation. For example, in addition, the identity element is 0, and in multiplication, it's 1.
3. The distributive property combines addition and multiplication and states that [tex]\(a \cdot (b + c) = a \cdot b + a \cdot c\)[/tex].
4. The associative property states that the way the operands are grouped does not affect the result. For addition, [tex]\((a + b) + c = a + (b + c)\)[/tex], and for multiplication, [tex]\((a \cdot b) \cdot c = a \cdot (b \cdot c)\)[/tex].
Given [tex]\(x \cdot y = y \cdot x\)[/tex], we can recognize this as an example of the commutative property because it directly describes the equality where changing the order of multiplication does not alter the result. Thus, this illustrates the commutative property.
Therefore, the answer is:
commutative property.
This property confirms that the order in which two complex numbers are multiplied does not change the product.
