To solve the given expression involving indices:

$\frac{7^{-8} \times 7^{-7}}{7^{-4}}$

we can use the laws of indices (also known as exponent rules).

Step 1: Multiply the terms in the numerator

When multiplying terms with the same base, we add their exponents. Using the rule:

$a^m \times a^n = a^{m+n}$

we have:

$7^{-8} \times 7^{-7} = 7^{-8 + (-7)} = 7^{-15}$

Step 2: Simplify the fraction

Now we have:

$\frac{7^{-15}}{7^{-4}}$

When dividing terms with the same base, we subtract their exponents. Using the rule:

$\frac{a^m}{a^n} = a^{m-n}$

we get:

$7^{-15 - (-4)} = 7^{-15 + 4} = 7^{-11}$

Final Answer:

The simplified expression is:

$7^{-11}$

Would you like more details on this solution, or any related questions?

Here are some questions that expand on this topic:

1. How would the result change if the base was different, for example, 5 instead of 7?
2. Can you simplify an expression like $\frac{7^3 \times 7^{-5}}{7^{-2}}$?
3. What happens if the exponents are fractions instead of integers?
4. How would you solve the problem if the denominator had a positive exponent?
5. What are some real-world applications of using indices in equations?

Tip: Remember, whenever dealing with indices, always apply the relevant rules of exponents, such as multiplication, division, and powers of powers, carefully to avoid mistakes.