To simplify the expression \(\frac{2^{2}x^{-1}y^{-8}}{9^{-1}x^{0}y^{-8}}\), we will follow these steps, addressing each component separately:
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Simplify the components:
- The base \(9^{-1}\) can be rewritten as \(\frac{1}{9}\), which is the same as multiplying by \(9\) in the denominator.
- The expression \(x^{0}\) equals \(1\) (any non-zero base raised to the power of zero equals one).
- The exponent \(y^{-8}\) appears in both the numerator and denominator, allowing it to cancel out.
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Rewrite the expression: \[ \frac{2^{2}x^{-1}y^{-8}}{9^{-1}x^{0}y^{-8}} = \frac{2^{2}x^{-1}y^{-8}}{\frac{1}{9} \cdot 1 \cdot y^{-8}} \]
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Simplify the division: Since dividing by a fraction is equivalent to multiplying by its reciprocal, we can rewrite the expression as: \[ 2^{2} x^{-1} y^{-8} \cdot 9 \cdot y^{8} \]
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Combine like terms: The \(y^{-8} \cdot y^{8}\) combines to \(y^{0}\) which equals \(1\) and thus can be eliminated. Now we have: \[ 2^{2} \cdot 9 \cdot x^{-1} \]
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Calculate the numerical components: Calculate \(2^2 = 4\), so the expression becomes: \[ 4 \cdot 9 \cdot x^{-1} = 36 \cdot x^{-1} \]
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Express with positive exponents: Finally, to express \(x^{-1}\) with a positive exponent, we write \(x^{-1} = \frac{1}{x}\): \[ 36 \cdot \frac{1}{x} = \frac{36}{x} \]
Thus, the simplified expression using only positive exponents is: \[ \frac{36}{x} \]