To determine which properties of multiplication are used to rewrite the expression [tex]\(\left(\frac{2}{3} \cdot \frac{1}{5}\right) \cdot \frac{5}{2}\)[/tex] as [tex]\(\frac{5}{2} \cdot \left(\frac{1}{5} \cdot \frac{2}{3}\right)\)[/tex], we need to examine the steps involved:
1. Examine the Initial Expression:
[tex]\[
\left(\frac{2}{3} \cdot \frac{1}{5}\right) \cdot \frac{5}{2}
\][/tex]
2. Apply Commutative Property (First Step):
The commutative property of multiplication states that [tex]\(a \cdot b = b \cdot a\)[/tex]. Therefore, we can change the order of multiplication within each group:
[tex]\[
\frac{5}{2} \cdot \left(\frac{1}{5} \cdot \frac{2}{3}\right)
\][/tex]
3. Apply Associative Property (Second Step):
The associative property of multiplication states that [tex]\((a \cdot b) \cdot c = a \cdot (b \cdot c)\)[/tex]. Here’s how it works:
[tex]\[
\frac{5}{2} \cdot \left(\frac{1}{5} \cdot \frac{2}{3}\right)
\][/tex]
In summary, by first applying the commutative property to change the order of multiplication and then applying the associative property to regroup the factors, we can rewrite the expression as desired.
Therefore, the correct sequence of properties used is:
[tex]\[
\text{The commutative property followed by the associative property.}
\][/tex]
Thus, the correct option is:
[tex]\[
\boxed{\text{The commutative property followed by the associative property}}
\][/tex]
