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The expression you uploaded is:

\[
\frac{4^3 \cdot 9^7}{27^5 \cdot 8^2}
\]

We can simplify this step-by-step.

### Step 1: Express all terms with prime bases
We rewrite each base in terms of prime factors:
- \( 4 = 2^2 \), so \( 4^3 = (2^2)^3 = 2^6 \)
- \( 9 = 3^2 \), so \( 9^7 = (3^2)^7 = 3^{14} \)
- \( 27 = 3^3 \), so \( 27^5 = (3^3)^5 = 3^{15} \)
- \( 8 = 2^3 \), so \( 8^2 = (2^3)^2 = 2^6 \)

Thus, the expression becomes:

\[
\frac{2^6 \cdot 3^{14}}{3^{15} \cdot 2^6}
\]

### Step 2: Simplify the expression
Now cancel out common terms in the numerator and denominator:
- \( 2^6 \) appears in both the numerator and denominator, so they cancel out.
- For the powers of 3, subtract the exponents: \( 3^{14} - 3^{15} = 3^{-1} = \frac{1}{3} \).

Thus, the simplified expression is:

\[
\frac{1}{3}
\]

### Final Answer:
The value of the expression is \( \frac{1}{3} \).

Would you like any more details or have any questions? 

Here are five related questions:
1. How would the expression change if the bases were not powers of primes?
2. Can you simplify an expression like \( \frac{6^5 \cdot 12^3}{3^2 \cdot 8^4} \)?
3. What happens if the powers of two or three were different in this problem?
4. How do you deal with negative exponents in the numerator or denominator?
5. What role do the laws of exponents play in simplifying expressions?

**Tip:** Always look for ways to express terms with common prime factors when simplifying complex expressions.

Simplify the expression \( \frac{4^3 \cdot 9^7}{27^5 \cdot 8^2} \)

Learn how to simplify the expression \( \frac{4^3 \cdot 9^7}{27^5 \cdot 8^2} \) using the laws of exponents and prime factorization. This detailed step-by-step solution is suitable for students in Grades 9-10, covering key algebraic concepts of exponentiation.

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