The expression to simplify is:

$\frac{3(9 - 3)^4 - 6 \cdot 9 + 7 \cdot 3}{8^3 + 5^2}$

Step-by-Step Simplification

1. Simplify inside the parentheses: $9 - 3 = 6$

2. Calculate the exponentiation: $(6)^4 = 1296$

   So, the expression becomes: $\frac{3 \cdot 1296 - 6 \cdot 9 + 7 \cdot 3}{8^3 + 5^2}$

3. Compute the multiplication in the numerator: $3 \cdot 1296 = 3888, \quad 6 \cdot 9 = 54, \quad 7 \cdot 3 = 21$

   Substitute these values in: $\frac{3888 - 54 + 21}{8^3 + 5^2}$

4. Simplify the numerator: $3888 - 54 = 3834, \quad 3834 + 21 = 3855$

   So, the numerator simplifies to 3855.

5. Calculate the denominator: $8^3 = 512, \quad 5^2 = 25$

   $512 + 25 = 537$

6. Final expression: $\frac{3855}{537}$

7. Simplify the fraction:

   Divide both the numerator and the denominator by their greatest common divisor (GCD), which is 3: $\frac{3855 \div 3}{537 \div 3} = \frac{1285}{179}$

Final Answer

The simplified expression is:

$\frac{1285}{179}$

Would you like more details on any of the steps, or do you have any questions?

Additional Related Questions:

1. How do you find the greatest common divisor (GCD) of two numbers?
2. What are the rules for simplifying algebraic expressions with exponents?
3. How can you determine if a fraction can be reduced further?
4. What are some examples of expressions with different bases and exponents?
5. How can you handle expressions that include both multiplication and division?

Tip:

When simplifying expressions, always follow the order of operations: parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right).