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!

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.