To express the given expression as $\frac{2^m}{3^n}$, let's start by simplifying each term.

The given expression is:

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

Step 1: Simplify Each Term

1. $8 \left(\frac{2}{3}\right)^4$ $= 8 \cdot \frac{2^4}{3^4} = \frac{8 \cdot 2^4}{3^4} = \frac{2^7}{3^4}$

2. $-4 \left(\frac{2}{3}\right)^5$ $= -4 \cdot \frac{2^5}{3^5} = -\frac{4 \cdot 2^5}{3^5} = -\frac{2^7}{3^5}$

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

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

Step 2: Combine the Terms

Now, the expression becomes: $\frac{2^7}{3^4} - \frac{2^7}{3^5} + \frac{2^7}{3^6} + \frac{2^7}{3^6}$

Factor out $\frac{2^7}{3^6}$ from each term: $\frac{2^7}{3^4} - \frac{2^7}{3^5} + \frac{2^7}{3^6} + \frac{2^7}{3^6} = \frac{2^7}{3^6} \left(\frac{3^2}{1} - \frac{3^1}{3^1} + 1 + 1\right)$

Simplify inside the parentheses: $\frac{2^7}{3^6} \left(9 - 3 + 1 + 1\right) = \frac{2^7}{3^6} \cdot 8$

Now multiply: $= \frac{2^7 \cdot 2^3}{3^6} = \frac{2^{10}}{3^6}$

Thus, the expression simplifies to:

$\frac{2^{10}}{3^6}$

Final Answer

The expression in the form $\frac{2^m}{3^n}$ is: $\frac{2^{10}}{3^6}$ where $m = 10$ and $n = 6$.

Would you like any further details or have any questions? Here are some related questions you might be interested in:

1. How do you factor out common terms in algebraic expressions?
2. What is the importance of exponents in mathematical expressions?
3. How do you simplify complex fractions?
4. What is the significance of expressing equations in the form $\frac{a^m}{b^n}$?
5. How can you use algebraic identities to simplify expressions?

Tip: When simplifying expressions, always look for common factors to factor out, which can make the problem much easier to handle.