To determine which expression correctly applies the Product Rule of Exponents to simplify to 2412, we need to recall the Product Rule. The Product Rule states that when multiplying two powers with the same base, you add their exponents. Mathematically, this is expressed as:
\[ a^m \cdot a^n = a^{m+n} \]
To arrive at a result of 2412, we need to consider possible bases and exponents that would yield that outcome when using the Product Rule.
Let's look for combinations where: \[ a^{m+n} = 2412 \]
To find suitable expressions, you might consider factoring 2412 and expressing it in terms of powers. For instance, let's check if 2412 can be broken down into components based on the Product Rule:
If we find that: \[ a^3 \cdot a^4 = a^{3+4} = a^7 \] and \[ a^2 \cdot a^5 = a^{2+5} = a^7, \] then any evaluated base for \(a\) (like 2, 3, etc.) leading to 2412 could be used depending on correct prior manipulation.
Alternatively, without loss of generality, if you had specific expressions to evaluate, and they lead to \(2412\) via the Product Rule, then one of those could be selected.
For clarity, here are a few responses that would simplify correctly to 2412 based on possible factors:
- \(6^3 \cdot 6^2 = 6^{5} = 7776\) (not relevant)
- \(3^4 \cdot 3^2 = 3^{6} = 729\) (not relevant)
If you share specific expression options, I'd be happy to confirm which aligns with the Product Rule yielding a result of 2412. Otherwise, keep in mind we need to check for viable \(6402, 4603\), or simpler bases contributing effectively.
However, without specific options to choose from, consider revisiting inputs making \(2412\) a derived target through simpler exponentials that reach the requirement succinctly.