To simplify \( 81^{-3/4} \), we can follow these steps:
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Rewrite 81 as a power of 3: \[ 81 = 3^4 \] Thus, we have: \[ 81^{-3/4} = (3^4)^{-3/4} \]
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Use the power of a power rule, which states \((a^m)^n = a^{m \cdot n}\): \[ (3^4)^{-3/4} = 3^{4 \cdot (-3/4)} \]
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Simplify the exponent: \[ 4 \cdot (-3/4) = -3 \] Therefore: \[ 3^{4 \cdot (-3/4)} = 3^{-3} \]
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Rewrite \( 3^{-3} \): \[ 3^{-3} = \frac{1}{3^3} \]
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Compute \( 3^3 \): \[ 3^3 = 27 \]
Finally, we have: \[ 81^{-3/4} = \frac{1}{27} \]
Thus, the simplified form of \( 81^{-3/4} \) is: \[ \boxed{\frac{1}{27}} \]