To determine which expression simplifies to include a negative exponent, let's analyze each option one by one.
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Expression: \( \frac{2^{-12}}{2^{-6}} \)
Apply the quotient rule of exponents: \[ \frac{a^m}{a^n} = a^{m-n} \] \[ \frac{2^{-12}}{2^{-6}} = 2^{-12 - (-6)} = 2^{-12 + 6} = 2^{-6} \] This expression simplifies to \( 2^{-6} \), which is indeed a negative exponent.
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Expression: \( 2^5 \cdot 2^{-2} \)
Apply the product rule of exponents: \[ a^m \cdot a^n = a^{m+n} \] \[ 2^5 \cdot 2^{-2} = 2^{5 + (-2)} = 2^{5 - 2} = 2^3 \] This simplifies to \( 2^3 \), which is a positive exponent.
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Expression: \( -2(-3)^2 \)
Simplifying the expression: \[ -2(-3)^2 = -2 \cdot 9 = -18 \] This does not involve exponents at all and results in a negative value but not a negative exponent.
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Expression: \( 2^3 - 2^4 \)
This expression does not involve any exponents being multiplied or divided: \[ 2^3 - 2^4 = 8 - 16 = -8 \] While the calculation results in a negative number, it does not produce a negative exponent.
After reviewing all options, the correct expression that results in a negative exponent when simplified is:
\( \frac{2^{-12}}{2^{-6}} = 2^{-6} \).
1 answer