To use the Quotient Property of exponents, we start with the conversion of the mixed numbers \(9 \frac{3}{5}\) and \(9 \frac{1}{5}\) into improper fractions.
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Convert mixed numbers to improper fractions: \[ 9 \frac{3}{5} = 9 + \frac{3}{5} = \frac{45}{5} + \frac{3}{5} = \frac{48}{5} \] \[ 9 \frac{1}{5} = 9 + \frac{1}{5} = \frac{45}{5} + \frac{1}{5} = \frac{46}{5} \]
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Applying the Quotient Property of Exponents: The Quotient Property states: \[ a^m \div a^n = a^{m-n} \] In our case: \[ \frac{(9 \frac{3}{5})}{(9 \frac{1}{5})} = \frac{\frac{48}{5}}{\frac{46}{5}} = \frac{48}{46} \]
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Simplify the fraction: \[ \frac{48}{46} = \frac{24}{23} \]
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Using the Quotient Property of Exponents: If we express \(9\) with an exponent: \[ 9^{\frac{3}{5}} \div 9^{\frac{1}{5}} = 9^{\frac{3}{5} - \frac{1}{5}} = 9^{\frac{2}{5}} \]
Thus, the simplified exponent in fraction form is: \[ \frac{2}{5} \]
So, the final answer is: The simplified exponent is \( \frac{2}{5} \).
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