Question

What are the correct steps to simplify an exponential expression using the quotient rule of integer exponents?

Answer 1

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

1. **Understand the Quotient Rule**: The quotient rule states that for any non-zero numbers \( a \) and \( b \), and any integers \( m \) and \( n \):
\[
\frac{a^m}{a^n} = a^{m-n}
\]
This means that when you divide two exponential expressions with the same base, you subtract the exponent of the denominator from the exponent of the numerator.

2. **Identify the bases**: Ensure that both the numerator and denominator have the same base. If they do not share the same base, the quotient rule cannot be applied directly.

3. **Subtract the Exponents**: If the bases are the same, apply the quotient rule by subtracting the exponent in the denominator from the exponent in the numerator.
\[
\frac{a^m}{a^n} = a^{m-n}
\]

4. **Simplify the Result**: After applying the quotient rule, simplify the resulting expression if possible. This may include reducing any coefficients or further simplifying any remaining exponents.

5. **Check and Adjust for Negative Exponents**: If your simplification results in a negative exponent, remember that:
\[
a^{-k} = \frac{1}{a^k}
\]
for any integer \( k \), where \( a \neq 0 \).

### Example:

Suppose you want to simplify the expression \(\frac{x^5}{x^2}\).

1. Check the bases: Both are \( x \).
2. Apply the quotient rule:
\[
\frac{x^5}{x^2} = x^{5-2}
\]
3. Simplify the exponent:
\[
x^{5-2} = x^3
\]
4. The final result is \( x^3 \).

By following these steps, you can systematically simplify exponential expressions using the quotient rule of integer exponents.