Question: 5. Simplify $\frac{7^n}{7^n - 7^{n-1}}$
1. **State the problem:** Simplify the expression $$\frac{7^n}{7^n - 7^{n-1}}$$.
2. **Recall the properties of exponents:**
- $a^m \div a^n = a^{m-n}$
- $a^m - a^n$ can sometimes be factored by taking out the common factor.
3. **Rewrite the denominator by factoring out the common term:**
$$7^n - 7^{n-1} = 7^{n-1}(7 - 1)$$
4. **Substitute back into the expression:**
$$\frac{7^n}{7^{n-1}(7 - 1)}$$
5. **Simplify the numerator using exponent rules:**
$$7^n = 7^{n-1} \times 7$$
6. **Rewrite the fraction:**
$$\frac{7^{n-1} \times 7}{7^{n-1} \times (7 - 1)}$$
7. **Cancel the common factor $7^{n-1}$:**
$$\frac{\cancel{7^{n-1}} \times 7}{\cancel{7^{n-1}} \times (7 - 1)} = \frac{7}{7 - 1}$$
8. **Calculate the denominator:**
$$7 - 1 = 6$$
9. **Final simplified expression:**
$$\frac{7}{6}$$
**Answer:** The simplified form of $$\frac{7^n}{7^n - 7^{n-1}}$$ is $$\frac{7}{6}$$.