Simplify (12×4)⁴ ÷ (4×12)¹¹: Complex Exponential Fraction Problem

To solve this problem, we will use the power of a quotient rule for exponents, which states: $\frac{a^m}{a^n} = a^{m-n}$.

Here's a step-by-step solution:

• Step 1: Identify the base and exponents in the problem.
• Step 2: Apply the exponent rules.
• Step 3: Simplify the expression by calculating the difference in exponents.

Now, let's apply these steps in detail:

Step 1: We have the expression $\frac{(12 \times 4)^4}{(4 \times 12)^{11}}$. Here, the base is $(12 \times 4)$, and the exponents are 4 in the numerator and 11 in the denominator.

Step 2: Combine the terms using the commutative and associative properties of multiplication. Notice that the terms $12 \times 4$ are identical in both the numerator and denominator. So, simplify using these instances: $(12 \times 4)^4$ in the numerator and $(12 \times 4)^{11}$ in the denominator.

Step 3: Apply the power of a quotient rule:
$\frac{(12 \times 4)^4}{(12 \times 4)^{11}} = (12 \times 4)^{4-11} = (12 \times 4)^{-7}$

This means the simplified expression is $(12 \times 4)^{-7}$.

Therefore, the correct answer among the provided choices is $\left(12\times4\right)^{-7}$, which corresponds to choice 1.