Simplify the Fraction: 1/(3⁴×12⁴) Expression Challenge

To address this mathematical problem, we need to rewrite the expression $\frac{1}{3^4 \times 12^4}$ using exponent properties. Specifically, we'll use the property that tells us $\frac{1}{a^n} = a^{-n}$.

Let's perform the necessary steps:

• Step 1: Apply the formula to each component in the denominator. For $\frac{1}{3^4}$, we have $3^{-4}$ using the rule $\frac{1}{a^n} = a^{-n}$.
• Step 2: Similarly, for $\frac{1}{12^4}$, we apply the rule to get $12^{-4}$.
• Step 3: Combine these results using multiplication: $3^{-4} \times 12^{-4}$.

By performing these transformations, we can confirm that the expression $\frac{1}{3^4 \times 12^4}$ is equivalent to $3^{-4} \times 12^{-4}$.

Therefore, the correct expression is $3^{-4} \times 12^{-4}$, which is choice 3 in the multiple-choice options provided.