We are given the expression:

$\frac{7^{2n+5} - 7^{2n+4}}{7^{2n+4}}$

To simplify this, we can factor out $7^{2n+4}$ from the terms in the numerator. Here's the step-by-step process:

Step 1: Factor the numerator

Notice that $7^{2n+5}$ can be written as $7 \times 7^{2n+4}$. So, we factor out $7^{2n+4}$ from both terms in the numerator:

$7^{2n+5} - 7^{2n+4} = 7 \times 7^{2n+4} - 7^{2n+4} = 7^{2n+4}(7 - 1)$

Thus, the expression becomes:

$\frac{7^{2n+4}(7 - 1)}{7^{2n+4}}$

Step 2: Simplify the expression

Since $7^{2n+4}$ appears in both the numerator and denominator, they cancel each other out:

$7 - 1 = 6$

So, the simplified expression is:

$6$

Final Answer:

The simplified result is $6$.

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Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How would the expression change if the base were something other than 7?
2. What happens if we use a different exponent structure in the original problem?
3. How do we simplify expressions involving exponential terms in the denominator and numerator?
4. Can you generalize this method to an expression like $\frac{a^{2n+5} - a^{2n+4}}{a^{2n+4}}$?
5. What if there were additional terms in the numerator?

Tip: When simplifying expressions involving exponents, always check for common factors to factor out and simplify the expression efficiently.