To determine which properties can be 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 analyze the properties of arithmetic operations at play here:
### Step-by-Step Analysis:
1. Starting Expression:
[tex]\[
\left( \frac{2}{3} \cdot \frac{1}{5} \right) \cdot \frac{5}{2}
\][/tex]
2. Understanding the Target Expression:
We want to rewrite it as:
[tex]\[
\frac{5}{2} \cdot \left( \frac{1}{5} \cdot \frac{2}{3} \right)
\][/tex]
3. Identify Necessary Properties:
- Commutative Property of Multiplication: This property states that the order of multiplication does not affect the result:
[tex]\[
a \cdot b = b \cdot a
\][/tex]
- Associative Property of Multiplication: This property states that the grouping of multiplication does not affect the result:
[tex]\[
(a \cdot b) \cdot c = a \cdot (b \cdot c)
\][/tex]
4. Applying Properties:
- Apply the associative property first to rewrite the inner product:
[tex]\[
\left( \frac{2}{3} \cdot \frac{1}{5} \right) \cdot \frac{5}{2} = \frac{2}{3} \cdot \left( \frac{1}{5} \cdot \frac{5}{2} \right)
\][/tex]
- Apply the commutative property to rearrange the product within the parentheses:
[tex]\[
\frac{2}{3} \cdot \left( \frac{5}{2} \cdot \frac{1}{5} \right)
\][/tex]
- Finally, apply the commutative property again to move [tex]\(\frac{2}{3}\)[/tex] to the other side:
[tex]\[
\frac{5}{2} \cdot \left( \frac{1}{5} \cdot \frac{2}{3} \right)
\][/tex]
### Conclusion:
To rewrite the given expression, we first used the associative property once and then used the commutative property within the new grouping. Hence, the appropriate method was using the associative property used once.
So, the correct answer to the question is:
- The associative property used once
