To simplify the expression \( \frac{23^{65}}{23^{32}} \) using the quotient rule of integer exponents, we use the rule which states that for any non-zero base \( a \) and integers \( m \) and \( n \):
\[ \frac{a^m}{a^n} = a^{m-n} \]
In this case, we have:
- \( a = 23 \)
- \( m = 65 \)
- \( n = 32 \)
Applying the quotient rule:
\[ \frac{23^{65}}{23^{32}} = 23^{65 - 32} \]
Now, calculate \( 65 - 32 \):
\[ 65 - 32 = 33 \]
Thus, we have:
\[ \frac{23^{65}}{23^{32}} = 23^{33} \]
So the simplified expression is:
\[ 23^{33} \]