Let's answer the given question step-by-step.
We start with the original expression:
[tex]\[ \left(\frac{1}{3} \cdot \frac{2}{3}\right) \cdot \frac{5}{2} = \frac{2}{3} \cdot\left(\frac{1}{3} \cdot \frac{5}{2}\right) \][/tex]
The properties of multiplication we need to consider are:
1. The commutative property of multiplication, which states that [tex]\(a \cdot b = b \cdot a\)[/tex].
2. The associative property of multiplication, which states that [tex]\((a \cdot b) \cdot c = a \cdot (b \cdot c)\)[/tex].
Let's examine the transformation of the left-hand side (LHS) of the expression to the right-hand side (RHS).
- Initially, on the LHS, the expression is grouped as [tex]\(\left(\frac{1}{3} \cdot \frac{2}{3}\right) \cdot \frac{5}{2}\)[/tex].
- On the RHS, the expression is grouped as [tex]\(\frac{2}{3} \cdot \left(\frac{1}{3} \cdot \frac{5}{2}\right)\)[/tex].
To transform LHS to RHS:
- We see that we need to regroup the factors in the multiplication.
- This regrouping uses the associative property of multiplication. Applying the associative property, we change the grouping while maintaining the order of multiplication.
Thus, the correct and specific property being used in this transformation is the associative property used once.
So, the answer is:
[tex]\[ \textbf{The associative property used once.} \][/tex]
Next, we are asked to rewrite [tex]\(35 \cdot y\)[/tex] in a different way using the commutative property of multiplication. According to the commutative property:
[tex]\[ 35 \cdot y = y \cdot 35 \][/tex]
Hence, using the commutative property, [tex]\(35 \cdot y\)[/tex] can be rewritten as [tex]\(y \cdot 35\)[/tex].
