To simplify the expression \( 35^{3} \cdot 35^{-9} \) to \( 35^{14} \), we can use the Product Rule of Exponents, which states that when multiplying two expressions with the same base, you add the exponents:
\[ a^m \cdot a^n = a^{m+n} \]
In this case:
\[ 35^{3} \cdot 35^{-9} = 35^{3 + (-9)} = 35^{-6} \]
If you wanted to express \( 35^{-6} \) as a positive exponent, you'd apply the Property of Negative Exponents, which states that:
\[ a^{-m} = \frac{1}{a^{m}} \]
Thus, \( 35^{-6} = \frac{1}{35^{6}} \).
So, the properties applied to generate the equivalent expression \( 35^{-6} \) from \( 35^{3} \cdot 35^{-9} \) are the Property of Negative Exponents and the Product Rule of Exponents.
Therefore, the correct response is:
the Property of Negative Exponents and the Product Rule of Exponents.
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