Final answer:
Yes, Alana is correct; (ab)²(ab) and [(ab)]² represent the same product due to associative and commutative properties of multiplication.
Step-by-step explanation:
Alana is correct that (ab)²(ab) is the same as [(ab)]². This is because of the associative property of multiplication, which allows us to group factors in different ways without changing the product. In this case, (ab)² means ab multiplied by itself, or (ab)(ab), and then this result is multiplied by another ab, giving us (ab)(ab)(ab). When we denote [(ab)]², it implies first calculating (ab) and then squaring the result, which mathematically also translates to (ab)(ab) and finally multiplying by another ab does not change the sequence or the product. The commutative property of multiplication further allows us to rearrange the factors without altering the product.