To simplify the expression \(\frac{2^{2}x^{-1}y^{-8}}{9^{-1}x^{0}y^{-8}}\), we will first rewrite \(9^{-1}\) and \(x^{0}\):
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Recognize that \(9^{-1} = \frac{1}{9}\) and rewrite \(9\) as \(3^2\). Thus, \[ 9^{-1} = \frac{1}{3^2} = 3^{-2} \] Therefore, we can rewrite the expression as: \[ \frac{2^{2}x^{-1}y^{-8}}{3^{-2}x^{0}y^{-8}} \]
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Since \(x^{0} = 1\), it simplifies to: \[ \frac{2^{2}x^{-1}y^{-8}}{3^{-2}y^{-8}} = \frac{2^{2}x^{-1}}{3^{-2}} \]
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Now, multiply both the numerator and denominator by \(3^{2}\) to eliminate the negative exponent in the denominator: \[ 2^{2}x^{-1} \cdot 3^{2} = 2^{2} \cdot 3^{2} \cdot x^{-1} \]
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Using the positive exponent property on \(x^{-1}\), we can write: \[ x^{-1} = \frac{1}{x} \] Therefore, the expression becomes: \[ \frac{2^{2} \cdot 3^{2}}{x} = \frac{4 \cdot 9}{x} = \frac{36}{x} \]
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Since we need to write the final answer using only positive exponents, we get: \[ 36x^{-1} \]
Putting it all together, the simplified form of the given expression is: \[ \frac{36}{x} \]
So, the final answer is: \[ \frac{36}{x} \] or \(36x^{-1}\) (both represent the same value).