To determine which expression simplifies to a negative exponent, let's simplify each option:
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\(33−34\): \[ 3^3 - 3^4 = 27 - 81 = -54 \] This results in a negative number, but not an exponent.
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\( \frac{3^3}{3^7} \): Using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \): \[ \frac{3^3}{3^7} = 3^{3-7} = 3^{-4} \] This results in a negative exponent.
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\((-3)^2\): \[ (-3)^2 = 9 \] This is a positive number.
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\(3^5 \cdot (-3)^2\): First, compute \((-3)^2\): \[ (-3)^2 = 9 \rightarrow 3^5 \cdot 9 = 3^5 \cdot 3^2 = 3^{5+2} = 3^7 \] This results in a positive exponent.
Thus, the only expression that simplifies to a negative exponent is:
\( \frac{3^3}{3^7} \) (the fraction with a numerator of \( 3^3 \) and a denominator of \( 3^7 \)).