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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.

Simplify the expression: (7^-8 * 7^-7) / 7^-4

Learn how to simplify the expression (7^-8 * 7^-7) / 7^-4 using the laws of indices. This algebraic problem demonstrates how to apply exponent rules for multiplication and division of powers. Suitable for students in Grades 9-10.

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