 8 { ( \dfrac{ 2  }{ 3  }  )  }^{ 4  }  -4 { ( \dfrac{ 2  }{ 3  }  )  }^{ 5  }  +2 { ( \dfrac{ 2  }{ 3  }  )  }^{ 6  }  +6 { ( \dfrac{ 2  }{ 3  }  )  }^{ 7  }   express as 2^m/3^n

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.

Learn how to express the complex expression 8(2/3)^4 - 4(2/3)^5 + 2(2/3)^6 + 6(2/3)^7 as a fraction 2^m/3^n. This detailed solution covers exponentiation, fractional powers, and simplification techniques for algebraic expressions. Suitable for high school students studying advanced algebra.

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