Apply the Quotient Rule of Integer Exponents to simplify 23^65 divided 23^32

Answer 1

The Quotient Rule of Integer Exponents states that for any nonzero base \( a \) and integers \( m \) and \( n \):

\[ \frac{a^m}{a^n} = a^{m-n} \]

In your case, we have:

\[ \frac{23^{65}}{23^{32}} \]

Applying the Quotient Rule:

\[ \frac{23^{65}}{23^{32}} = 23^{65 - 32} = 23^{33} \]

So, the simplified expression is:

\[ 23^{33} \]

Answer 2

Apply the Quotient Rule of Integer Exponents to simplify 12^53 divided by 7^53

Answer 3

The Quotient Rule of Integer Exponents applies only to expressions with the same base. In your case, we have:

\[ \frac{12^{53}}{7^{53}} \]

Since the bases (12 and 7) are different, we can't directly apply the Quotient Rule. However, we can express it as follows:

\[ \frac{12^{53}}{7^{53}} = \left(\frac{12}{7}\right)^{53} \]

So, the simplified expression is:

\[ \left(\frac{12}{7}\right)^{53} \]