Question

Which are the correct steps to follow to simplify an expotential expression using the quotient role of integer exponents

Answer 1

To simplify an exponential expression using the quotient rule of integer exponents, follow these steps:

1. **Identify the Base and Exponents**: Ensure that you have a common base in the form \( \frac{a^m}{a^n} \), where \( a \) is the base, and \( m \) and \( n \) are the exponents.

2. **Apply the Quotient Rule**: According to the quotient rule of exponents, when you divide two powers with the same base, you subtract the exponent in the denominator from the exponent in the numerator:
\[
\frac{a^m}{a^n} = a^{m-n}
\]

3. **Simplify the Exponent**: Perform the subtraction \( m - n \) to simplify the exponent. The resulting expression will be:
\[
a^{m-n}
\]

4. **Finalize the Expression**: Rewrite the simplified expression. If \( m - n \) is a negative number, be aware that it can also be rewritten as:
\[
a^{m-n} = \frac{1}{a^{n-m}} \quad (\text{if } m < n)
\]

### Example:
Suppose you want to simplify \( \frac{x^5}{x^2} \).

1. Identify the base \( x \) and the exponents \( 5 \) and \( 2 \).
2. Apply the quotient rule:
\[
\frac{x^5}{x^2} = x^{5-2} = x^3
\]
3. Final expression: \( x^3 \)

By following these steps, you can simplify exponential expressions using the quotient rule effectively.