To simplify the expression \( \frac{23^{65}}{22^{32}} \) using the quotient rule of integer exponents, you start with the general rule which states that:
\[ \frac{a^m}{a^n} = a^{m-n} \]
However, in this case, the bases \( 23 \) and \( 22 \) are different, so we cannot directly apply the quotient rule to combine them.
Thus, we leave the expression in its original form. You can also express it in a different way to illustrate the division of distinct bases:
\[ \frac{23^{65}}{22^{32}} = 23^{65} \cdot \frac{1}{22^{32}} \]
or rewrite it as:
\[ \frac{23^{65}}{22^{32}} = \frac{23^{65}}{1} \cdot 22^{-32} \]
This means:
\[ 23^{65} \cdot 22^{-32} \]
In conclusion, while \( \frac{23^{65}}{22^{32}} \) cannot be simplified further using the quotient rule since the bases are different, it can be expressed with negative exponents:
\[ \frac{23^{65}}{22^{32}} = 23^{65} \cdot 22^{-32} \]
This gives a proper mathematical representation of the original expression.
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