Answer :
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], we need to use certain properties of multiplication for fractions. Here’s the detailed, step-by-step solution:
1. Associative Property of Multiplication:
- The associative property of multiplication states that the way in which factors are grouped does not affect the product. In other words, [tex]\( (a \cdot b) \cdot c = a \cdot (b \cdot c) \)[/tex].
- Start by applying this property to the original expression [tex]\(\left(\frac{2}{3} \cdot \frac{1}{5}\right) \cdot \frac{5}{2}\)[/tex]:
[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]
2. Commutative Property of Multiplication:
- The commutative property of multiplication states that the order of factors does not affect the product, i.e., [tex]\( a \cdot b = b \cdot a \)[/tex].
- Use this property to reorder the multiplication within the parentheses:
[tex]\[ \frac{2}{3} \cdot \left(\frac{1}{5} \cdot \frac{5}{2}\right) = \frac{2}{3} \cdot \left(\frac{5}{2} \cdot \frac{1}{5}\right) \][/tex]
Hence, to rewrite [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], we have used the associative property first to change the grouping and then the commutative property to change the order of multiplication.
Therefore, the correct answer is:
- The associative property followed by the commutative property
1. Associative Property of Multiplication:
- The associative property of multiplication states that the way in which factors are grouped does not affect the product. In other words, [tex]\( (a \cdot b) \cdot c = a \cdot (b \cdot c) \)[/tex].
- Start by applying this property to the original expression [tex]\(\left(\frac{2}{3} \cdot \frac{1}{5}\right) \cdot \frac{5}{2}\)[/tex]:
[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]
2. Commutative Property of Multiplication:
- The commutative property of multiplication states that the order of factors does not affect the product, i.e., [tex]\( a \cdot b = b \cdot a \)[/tex].
- Use this property to reorder the multiplication within the parentheses:
[tex]\[ \frac{2}{3} \cdot \left(\frac{1}{5} \cdot \frac{5}{2}\right) = \frac{2}{3} \cdot \left(\frac{5}{2} \cdot \frac{1}{5}\right) \][/tex]
Hence, to rewrite [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], we have used the associative property first to change the grouping and then the commutative property to change the order of multiplication.
Therefore, the correct answer is:
- The associative property followed by the commutative property
