To determine which equation exemplifies the commutative property of multiplication, let us first recall the definition of the commutative property. The commutative property of multiplication states that changing the order of the factors does not change the product. That is, for any two numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex], [tex]\(a \cdot b = b \cdot a\)[/tex].
Now, let's analyze each option in detail:
1. [tex]\((4 + 2i) = (2i + 4)\)[/tex]
- This equation illustrates the commutative property of addition, not multiplication. It shows that the addition of complex numbers does not depend on the order of the terms.
2. [tex]\((4 + 2i)(3 - 5i) = (3 - 5i)(4 + 2i)\)[/tex]
- This is the correct example of the commutative property of multiplication because it shows that the product remains the same regardless of the order in which the two complex numbers are multiplied.
3. [tex]\((4 + 2i)(3 - 5i) = (4 + 2i)(3 - 5i)(1)\)[/tex]
- This equation introduces an additional factor of [tex]\(1\)[/tex] on the right-hand side, which does not fit the commutative property of multiplication. The equation is actually illustrating the identity property of multiplication.
4. [tex]\((4 + 2i) = (4 + 2i + 0)\)[/tex]
- This equation exemplifies the identity property of addition, as it shows that adding [tex]\(0\)[/tex] to a number leaves it unchanged.
Based on this analysis, the equation that properly demonstrates the commutative property of multiplication is:
[tex]\[
(4 + 2i)(3 - 5i) = (3 - 5i)(4 + 2i)
\][/tex]
Therefore, the correct answer is the second option.
