Question

Which are the correct steps to follow 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 should follow these steps:

1. **Identify the Expression**: Ensure that you have a fraction involving exponential terms in the form \(\frac{a^m}{a^n}\), where \(a\) is the base, and \(m\) and \(n\) are integers (the exponents).

2. **Apply the Quotient Rule**: According to the quotient rule for exponents, when you divide two exponential expressions 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. **Perform the Subtraction**: Execute the subtraction \(m-n\) to find the new exponent.

4. **Rewrite the Expression**: Rewrite the simplified expression with the new exponent:
\[
a^{m-n}
\]

5. **Simplify Further if Necessary**: If \(m-n\) can be simplified further (e.g., if it’s a negative exponent), apply the rules for negative exponents if needed:
\[
a^{-k} = \frac{1}{a^k}
\]

### Example

For example, simplify \(\frac{x^5}{x^2}\):

1. **Identify the expression**: \(\frac{x^5}{x^2}\)
2. **Apply the quotient rule**: We apply \(x^{5-2}\).
3. **Perform the subtraction**: \(5 - 2 = 3\).
4. **Rewrite the expression**: This gives us \(x^3\).
5. **Simplify further if necessary**: In this case, no further simplification is needed.

The final simplified result is \(x^3\).

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