The expression in the image is:

$\frac{4^5 \cdot 3^5}{12^3}$

Step-by-Step Solution:

1. Simplify the terms:

   • First, rewrite $12^3$ as a product of prime factors: $12 = 2^2 \cdot 3$ So, $12^3 = (2^2 \cdot 3)^3 = 2^6 \cdot 3^3$

2. Rewriting the expression: The expression becomes: $\frac{4^5 \cdot 3^5}{2^6 \cdot 3^3}$

3. Simplify powers of 4:

   • Since $4 = 2^2$, we can rewrite $4^5$ as: $4^5 = (2^2)^5 = 2^{10}$ Now, the expression looks like this: $\frac{2^{10} \cdot 3^5}{2^6 \cdot 3^3}$

4. Simplifying the powers:

   • For the base 2 terms: $2^{10} / 2^6 = 2^{10 - 6} = 2^4$
   • For the base 3 terms: $3^5 / 3^3 = 3^{5 - 3} = 3^2$

   Now, the expression is: $2^4 \cdot 3^2$

5. Calculate the result:

   • $2^4 = 16$
   • $3^2 = 9$

   Therefore, the simplified expression is: $16 \cdot 9 = 144$

Final Answer:

The value of the expression is $144$.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. How do you simplify exponents when multiplying terms with the same base?
2. Can you apply this method to simplify other similar expressions?
3. What happens if the denominator had different bases? How would that change the solution process?
4. Can you solve expressions that involve negative exponents?
5. How do you handle division of exponents with fractional or decimal bases?

Tip: Always simplify exponents by breaking them down into prime factors whenever possible. This can make it easier to compare and reduce terms across the equation.