Renee is simplifying the expression (7) (StartFraction 13 over 29 EndFraction) (StartFraction 1 over 7 EndFraction). She recognizes that 7 and StartFraction 1 over 7 EndFraction are reciprocals, so she would like to find their product before she multiplies by StartFraction 13 over 29 EndFraction. Which property will allow Renee to do this without changing the value of the expression?

Question

associative property
commutative property
distributive property
identity property

Answer 1

Renee can use the associative property to regroup the factors in the expression without changing its value. The associative property states that the way in which numbers are grouped in multiplication does not change the product.

In her case, she can regroup \( (7) \) and \( \left( \frac{1}{7} \right) \) and multiply them first, since they are reciprocals. Thus, the expression can be rewritten as:

\[ (7 \cdot \frac{1}{7}) \cdot \frac{13}{29} \]

This regrouping allows her to simplify \( 7 \cdot \frac{1}{7} = 1 \), leading to a simpler calculation of \( 1 \cdot \frac{13}{29} = \frac{13}{29} \). So, the answer is the associative property.