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The given expression is:

\[
\left(\frac{7}{8}\right)^{-3} \times \left(\frac{-7}{8}\right)^{4} \div \left(\frac{4}{7}\right)^{-6}
\]

To simplify this expression, follow these steps:

### Step 1: Apply the negative exponents
- A negative exponent means you take the reciprocal of the base. So, applying the exponents:

\[
\left(\frac{7}{8}\right)^{-3} = \left(\frac{8}{7}\right)^{3}
\]

\[
\left(\frac{-7}{8}\right)^{4} = \frac{(-7)^{4}}{8^{4}} = \frac{2401}{4096} 
\]

\[
\left(\frac{4}{7}\right)^{-6} = \left(\frac{7}{4}\right)^{6}
\]

### Step 2: Rewrite the expression
The expression now becomes:

\[
\left(\frac{8}{7}\right)^{3} \times \frac{2401}{4096} \div \left(\frac{7}{4}\right)^{6}
\]

### Step 3: Convert division to multiplication
Dividing by a fraction is equivalent to multiplying by its reciprocal:

\[
\left(\frac{8}{7}\right)^{3} \times \frac{2401}{4096} \times \left(\frac{4}{7}\right)^{6}
\]

### Step 4: Combine the powers with the same base
Now, combine the fractions:

1. Combine the powers of \(\frac{8}{7}\):

\[
\left(\frac{8}{7}\right)^{3} \times \left(\frac{4}{7}\right)^{6}
\]

2. Simplify the powers:

\(\left(\frac{8}{7}\right)^3\) simplifies to \(\frac{512}{343}\)

\(\left(\frac{4}{7}\right)^6\) simplifies to \(\frac{4096}{117649}\)

Thus, you get:

\[\frac{512}{343} \times \frac{4096}{117649} \times \frac{2401}{4096}\]

Combining the fractions results in:

\[\frac{512 \times 2401}{343 \times 117649}\]

You can continue to multiply, but this is simplified.

If you have further queries or clarifications on this or any other math question, I’m here to help!

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Here are 5 related questions to practice:

1. How would you simplify \((\frac{5}{3})^{-2} \times (\frac{5}{3})^4\)?
2. What does it mean to have a negative exponent in a fraction?
3. Can you simplify the expression \((\frac{2}{5})^{-3} \times (\frac{5}{2})^3\)?
4. How do you multiply and divide fractions with different exponents?
5. What is the general rule for simplifying expressions with multiple exponents?

**Tip:** When dealing with exponents, always remember that negative exponents mean reciprocal, and fractional exponents mean roots.

Learn how to simplify complex expressions involving negative exponents and fractions. This example covers applying exponent rules, finding reciprocals, and combining powers. Suitable for high school students (Grades 9-12).

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