Let's analyze each step and identify the property used:

Line 1 to Line 2:

Original expression:
$(9x)(6y)$

Rewritten as:
$9 \cdot (x \cdot 6) \cdot y$

Property used: Associative Property of Multiplication
Explanation: The associative property states that for any numbers $a, b,$ and $c$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$. In this case, the multiplication order within parentheses was grouped differently. No actual rearrangement occurred yet; instead, the parentheses were simply shifted around to group $x \cdot 6$.

Line 2 to Line 3:

Expression changes from:
$9 \cdot (x \cdot 6) \cdot y$

to:
$9 \cdot (6 \cdot x) \cdot y$

Property used: Commutative Property of Multiplication
Explanation: The commutative property states that $a \cdot b = b \cdot a$. Here, $x \cdot 6$ was rewritten as $6 \cdot x$.

Line 3 to Line 4:

Expression changes from:
$9 \cdot (6 \cdot x) \cdot y$

to:
$(9 \cdot 6) \cdot (x \cdot y)$

Property used: Associative Property of Multiplication
Explanation: The expression was regrouped by associating $9 \cdot 6$ and $x \cdot y$ together. This is another example of using the associative property to change how terms are grouped.

Line 4 to Line 5:

Expression changes from:
$(9 \cdot 6)(x \cdot y)$

to:
$54xy$

Property used: Simplification (Multiplication)
Explanation: The multiplication inside the parentheses is computed, and $9 \cdot 6 = 54$. Then, the expression is simplified to combine the terms $54xy$.

Summary of Properties Used:

1. Line 1 to Line 2: Associative Property of Multiplication
2. Line 2 to Line 3: Commutative Property of Multiplication
3. Line 3 to Line 4: Associative Property of Multiplication
4. Line 4 to Line 5: Simplification (Multiplication)

Would you like more details on any of the properties?

Related Questions:

1. What is the difference between the commutative and associative properties?
2. How does the associative property help in simplifying expressions?
3. What other mathematical properties are commonly used in algebra?
4. Can you provide an example where the distributive property is used?
5. How would the steps change if there were subtraction instead of multiplication?

Tip: Always remember the order of operations (PEMDAS/BODMAS) when simplifying expressions.