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]\(\frac{2}{3} \cdot \left(\frac{5}{2} \cdot \frac{1}{5}\right)\)[/tex], let's explore the properties involved.
1. Commutative Property of Multiplication:
This property states that the order in which two numbers are multiplied does not affect the product. In other words, [tex]\(a \cdot b = b \cdot a\)[/tex].
2. Associative Property of Multiplication:
This property states that the way in which numbers are grouped in a multiplication problem does not affect the product. In other words, [tex]\((a \cdot b) \cdot c = a \cdot (b \cdot c)\)[/tex].
Let's start by examining the given expression [tex]\(\left(\frac{2}{3} \cdot \frac{1}{5}\right) \cdot \frac{5}{2}\)[/tex] and how we can manipulate it using these properties.
First, we'll apply the Commutative Property of Multiplication to change the order of [tex]\(\frac{1}{5}\)[/tex] and [tex]\(\frac{5}{2}\)[/tex]:
[tex]\[
\left(\frac{2}{3} \cdot \frac{1}{5}\right) \cdot \frac{5}{2} = \left(\frac{2}{3} \cdot \frac{5}{2}\right) \cdot \frac{1}{5}
\][/tex]
Next, we will observe that the new form can still be manipulated further. Apply the Associative Property of Multiplication to change the grouping:
[tex]\[
\left(\frac{2}{3} \cdot \frac{5}{2}\right) \cdot \frac{1}{5} = \frac{2}{3} \cdot \left(\frac{5}{2} \cdot \frac{1}{5}\right)
\][/tex]
To summarize, the expression [tex]\(\left(\frac{2}{3} \cdot \frac{1}{5}\right) \cdot \frac{5}{2}\)[/tex] has been rewritten as [tex]\(\frac{2}{3} \cdot \left(\frac{5}{2} \cdot \frac{1}{5}\right)\)[/tex]:
1. Using the Commutative Property to switch the order.
2. Applying the Associative Property to regroup the factors.
Therefore, the properties used to rewrite the expression are:
> The commutative property used twice.
