In the absence of complete information, step 5 of Darlene's proof involving the identity <(xy)^2 - (x-y)^2 = 4xy> could involve the Distributive Property if it includes expansion or combination of terms, or the Commutative Property if it simply involves rearranging terms.
To determine which property justifies step 5 in Darlene's proof that (xy)^2 - (x-y)^2 = 4xy, we need to consider how the equation has been manipulated from step 4 to step 5. Although the provided information is not complete, we can deduce that step 5 likely involves simplifying the equation further by using one of the properties of algebraic operations.
If step 5 includes breaking down a binomial square or rearranging the terms in a particular fashion, it would likely be either the Distributive Property if terms are being expanded or combined, or it could be the Commutative Property if terms are being reordered. The Associative Property refers to the grouping of terms and the Inverse Property relates to operations that reverse effects, such as addition and subtraction, or multiplication and division.
In many proofs involving the transformation of algebraic expressions, the Distributive Property is commonly used when expanding binomials or combining like terms. The property states that a(b + c) = ab + ac. If step 5 involves combining like terms or expanding parenthesis, then this would be the property being applied.
On the other hand, if step 5 involves simply rearranging terms without changing their values, without grouping or combining them differently, the Commutative Property (which allows the order of addition or multiplication to change, as in a + b = b + a or ab = ba) could be the correct justification. Without the explicit steps provided, an accurate identification of which property is used cannot be made.