Apply the quotient rule of integer exponents to simplify 23^65 divided by 22^32

Answer 1

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.