To determine which property is represented by the equation
[tex]\[
\frac{5}{2}\left(\frac{2}{7} \cdot \frac{2}{5}\right) = \frac{5}{2}\left(\frac{2}{5} \cdot \frac{2}{7}\right),
\][/tex]
let's carefully examine the equation.
The equation shows that the product inside the parentheses on both sides of the equation is the same, even though the order of the factors is swapped. To analyze this step-by-step:
1. Look at the left side of the equation:
[tex]\[
\frac{5}{2}\left(\frac{2}{7} \cdot \frac{2}{5}\right)
\][/tex]
2. On the right side:
[tex]\[
\frac{5}{2}\left(\frac{2}{5} \cdot \frac{2}{7}\right)
\][/tex]
Both sides involve the multiplication of fractions, but the order in which the fractions [tex]\(\frac{2}{7}\)[/tex] and [tex]\(\frac{2}{5}\)[/tex] are multiplied is reversed.
3. We can examine the general property of multiplication that allows us to swap the order of factors. This property states that for any numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex],
[tex]\[
a \cdot b = b \cdot a.
\][/tex]
This is known as the commutative property of multiplication. It indicates that the order of numbers being multiplied does not change the product.
So, the equation
[tex]\[
\frac{5}{2}\left(\frac{2}{7} \cdot \frac{2}{5}\right) = \frac{5}{2}\left(\frac{2}{5} \cdot \frac{2}{7}\right)
\][/tex]
demonstrates the commutative property of multiplication. Hence, the property represented by the given equation is:
[tex]\[
\text{commutative}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{1}
\][/tex]
