To simplify the expression \( 22x^{-1}y^{-89} - 1x^0y^{-8} \), we start by rewriting the terms using positive exponents.
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The term \( 22x^{-1}y^{-89} \) can be rewritten as: \[ \frac{22}{xy^{89}} \]
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The term \( 1x^0y^{-8} \) simplifies because \( x^0 = 1 \), so it becomes: \[ y^{-8} = \frac{1}{y^8} \]
Now we can rewrite this term: \[ 1x^0y^{-8} = \frac{1}{y^8} \]
So, our expression now looks like: \[ \frac{22}{xy^{89}} - \frac{1}{y^8} \]
To combine these two fractions, we need a common denominator, which would be \( xy^{89}y^8 = xy^{97} \).
Now we'll rewrite the two fractions:
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The first term becomes: \[ \frac{22}{xy^{89}} = \frac{22y^8}{xy^{97}} \]
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The second term becomes: \[ \frac{1}{y^8} = \frac{xy^{89}}{xy^{97}} \]
Now we combine them: \[ \frac{22y^8 - xy^{89}}{xy^{97}} \]
This would be the final simplified form of the expression: \[ \frac{22y^8 - xy^{89}}{xy^{97}} \]
Thus, the simplified expression with only positive exponents is: \[ \frac{22y^8 - xy^{89}}{xy^{97}} \]