https://file.aiquickdraw.com/mathbot/file/172651378782217f4teyj.jpg

In this image, we are dealing with a complex equation involving exponents and simplifying fractions. I will break down the explanation step by step:

### Initial Expression:
\[
\frac{4^x}{9^x} \cdot \frac{27^x \cdot 27}{8^x \cdot \frac{1}{8}} = \frac{2}{3}
\]

### Step-by-Step Explanation:

1. **Rewrite \(9^x\) and \(8^x\):**
   \[
   9^x = (3^2)^x = 3^{2x}, \quad 8^x = (2^3)^x = 2^{3x}
   \]
   This simplifies the first part of the equation as:
   \[
   \frac{4^x}{9^x} = \frac{2^{2x}}{3^{2x}}
   \]

2. **Simplifying the second part:**
   The second fraction can be simplified:
   \[
   \frac{27^x \cdot 27}{8^x \cdot \frac{1}{8}} = \frac{(3^3)^x \cdot 3^3}{2^{3x} \cdot \frac{1}{2^3}} = \frac{3^{3x+3}}{2^{3x-3}}
   \]

3. **Combining both parts:**
   Now, combining the two parts of the expression:
   \[
   \frac{2^{2x}}{3^{2x}} \cdot \frac{3^{3x+3}}{2^{3x-3}} = \frac{2^{2x} \cdot 3^{3x+3}}{3^{2x} \cdot 2^{3x-3}}
   \]

4. **Simplification of powers:**
   Simplify the powers of 2 and 3:
   \[
   \frac{2^{2x}}{2^{3x-3}} = 2^{2x - (3x-3)} = 2^{2x - 3x + 3} = 2^{-x+3}
   \]
   \[
   \frac{3^{3x+3}}{3^{2x}} = 3^{(3x+3) - 2x} = 3^{x+3}
   \]

5. **Final expression:**
   Now the simplified expression is:
   \[
   2^{-x+3} \cdot 3^{x+3} = \frac{2}{3}
   \]

   Since \(2^{-x+3} \cdot 3^{x+3}\) needs to equal \(\frac{2}{3}\), we can further simplify and equate the powers.

Let me know if you'd like additional details or further simplifications!

### Related Questions:
1. How do exponent rules apply when combining terms in a product?
2. What is the significance of simplifying \(9^x\) as \(3^{2x}\)?
3. How does the negative exponent affect the term \(2^{-x+3}\)?
4. Can this equation be generalized for any values of \(x\)?
5. How can we equate exponents for solving such equations in different contexts?

### Tip:
Always break down powers and exponents step by step, especially when simplifying expressions involving multiple terms raised to different powers.

Solve the algebraic equation involving exponents: \( \frac{4^x}{9^x} \cdot \frac{27^x \cdot 27}{8^x \cdot \frac{1}{8}} = \frac{2}{3} \)

Explore the step-by-step process of simplifying an algebraic equation with exponents, including powers of 2 and 3. This solution demonstrates the application of exponent rules, the simplification of fractional expressions, and key algebraic concepts. Suitable for high school students in Grades 10-12.

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