To simplify \( \frac{3(5)}{3(-9)} \) to arrive at \( 3(14) \), we can analyze how to handle the exponents:
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Power Rule of Exponents: This rule states that \( a^m \times a^n = a^{m+n} \). However, this is not directly applicable here since we are dealing with separate bases.
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Property of Negative Exponents: This property states that \( a^{-n} = \frac{1}{a^n} \). It's not applicable to the simplification in this case either.
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Product Rule of Exponents: This property states that \( a^m \cdot a^n = a^{m+n} \), which is not applicable in reaching from the fraction to the multiplied form directly because of separate bases.
In fact, the transformation from \( \frac{3(5)}{3(-9)} \) involves recognizing that the \( 3 \) in both the numerator and denominator can cancel out, a crucial step in the simplification process.
The best option that describes the situation given is:
the Property of Negative Exponents and the Product Rule of Exponents
This choice indicates that we notice the relationship and manipulate the bases accordingly while recognizing across the properties used.
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