To express the given expression in the form $\frac{2^m}{3^n}$, let's work through each term step by step.

Given Expression:

$8 \left( \frac{2}{3} \right)^4 - 4 \left( \frac{2}{3} \right)^5 + 2 \left( \frac{2}{3} \right)^6 + 6 \left( \frac{2}{3} \right)^7$

Step 1: Expand Each Term

1. $8 \left( \frac{2}{3} \right)^4$ $8 \cdot \frac{2^4}{3^4} = \frac{8 \cdot 16}{81} = \frac{128}{81}$

2. $-4 \left( \frac{2}{3} \right)^5$ $-4 \cdot \frac{2^5}{3^5} = -4 \cdot \frac{32}{243} = -\frac{128}{243}$

3. $2 \left( \frac{2}{3} \right)^6$ $2 \cdot \frac{2^6}{3^6} = 2 \cdot \frac{64}{729} = \frac{128}{729}$

4. $6 \left( \frac{2}{3} \right)^7$ $6 \cdot \frac{2^7}{3^7} = 6 \cdot \frac{128}{2187} = \frac{768}{2187}$

Step 2: Combine the Terms

Combine all these fractions together:

$\frac{128}{81} - \frac{128}{243} + \frac{128}{729} + \frac{768}{2187}$

Step 3: Express as $\frac{2^m}{3^n}$

Notice that each term has a factor of $2^7$ (or $128$), so we can factor out $\frac{128}{3^7}$ from the expression:

$\frac{128}{3^7} \left( \frac{2187}{81} - \frac{2187}{243} + \frac{2187}{729} + 6 \right)$

Now simplify the expression inside the parentheses:

$\frac{2187}{81} = 27, \quad \frac{2187}{243} = 9, \quad \frac{2187}{729} = 3$

$\text{Expression becomes: } 27 - 9 + 3 + 6 = 27$

Thus, the entire expression simplifies to:

$\frac{128 \cdot 27}{3^7} = \frac{128 \cdot 3^3}{3^7} = \frac{128}{3^{4}} = \frac{2^7}{3^4}$

Final Answer:

$\frac{2^7}{3^4}$

Summary:

The given expression simplifies to $\frac{2^7}{3^4}$.

Would you like more details or have any questions?

Here are five related questions for practice:

1. Simplify $16 \left(\frac{2}{5}\right)^3 - 4 \left(\frac{2}{5}\right)^4 + \left(\frac{2}{5}\right)^5$.
2. Express $3^6 \times 2^2 - 3^4 \times 2^4$ as a single power of $\frac{2^m}{3^n}$.
3. How would you simplify $\frac{2^8}{3^5} - \frac{2^7}{3^4} + \frac{2^6}{3^3}$?
4. Factor $x^4 - 4x^2 + 4$ completely.
5. Express $\left(\frac{3}{4}\right)^3 - \left(\frac{3}{4}\right)^5 + \left(\frac{3}{4}\right)^7$ as a single fraction.

Tip: When simplifying complex algebraic expressions, look for common factors that can be factored out to make the process easier.