Question

Apply the quotient rule of integer exponents to simplify 23 to the 65th power divided by 23 to the 32nd power

Answer 1

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:

\[
23^{33}
\]