Let's simplify the given expression step by step.

The expression is:

$\frac{7^{n+1} + 7^n}{7^n}$

Step 1: Factorize $7^{n+1}$

Using the exponent rule $a^{m+n} = a^m \cdot a^n$, we can rewrite $7^{n+1}$ as: $7^{n+1} = 7^n \cdot 7$

Substituting this into the numerator, the expression becomes: $\frac{7^n \cdot 7 + 7^n}{7^n}$

Step 2: Factorize the numerator

The numerator $7^n \cdot 7 + 7^n$ can be factored as: $7^n (7 + 1)$

So the expression becomes: $\frac{7^n (7 + 1)}{7^n}$

Step 3: Simplify

Cancel $7^n$ in the numerator and denominator: $7 + 1$

Final Answer:

$8$

Would you like further details or clarifications?

Questions for Deeper Understanding:

1. What rules of exponents are used in simplifying such expressions?
2. Can this method be applied to other bases or variables (e.g., $x^n + x^{n+1}$)?
3. Why does $7^n$ cancel when factoring the numerator?
4. What happens if $n$ were negative or zero?
5. How does the simplification change if additional terms are in the numerator?

Tip:

Always look for common factors in the numerator and denominator to simplify expressions efficiently.