To understand how \( 353 - 9 \) can be simplified to \( 314 \), we can use the properties of exponents.
The expression \( 353 - 9 \) can be interpreted as \( 35^{3} \cdot 35^{-9} \). We would use the Property of Negative Exponents to rewrite \( 35^{-9} \) as \( \frac{1}{35^9} \). However, when combining the exponents according to the Product Rule of Exponents, which states that \( a^m \cdot a^n = a^{m+n} \), we obtain:
\[ 35^{3} \cdot 35^{-9} = 35^{3 + (-9)} = 35^{-6} \]
This doesn't directly lead to \( 314 \), so let's assume that the question indeed meant \( 353^{-9} = 314 \).
To go from \( 353^{-9} \) directly to a simplified form, we would be applying the Property of Negative Exponents, \( a^{-n} = \frac{1}{a^n} \).
However, based on the options provided, the correct answer is:
the Property of Negative Exponents and the Product Rule of Exponents.
This choice covers both the use of negative exponents (for rewriting) as well as the handling of multiplication of powers.
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