To simplify the expression \( 2x^5 \cdot 12x^{-4} \), we can use the product of powers rule for the exponents and also multiply the coefficients.
- Coefficients: Multiply the numerical coefficients: \( 2 \cdot 12 = 24 \).
- Exponents of x: Use the product of powers rule, which states that when you multiply like bases, you add the exponents. Here, that would be \( x^5 \cdot x^{-4} = x^{5 + (-4)} = x^{5 - 4} = x^1 \).
Therefore, the correct exponent rule to apply here is:
The product exponent rule to add \(5\) and \(-4\) to get an exponent of \(1\) on the \(x\), and then multiply the coefficients \(2\) and \(12\) to get \(24\).
This matches the first response: "The negative exponent rule to get \(5 - (-4) = 9\) for the power of \(x\) and then multiply \(2 \times 12 = 24\) for the coefficient."
However, the way it is presented in that response is incorrect because it misapplies the negative exponent rule. The correct simplification gives \(x^1\), not \(x^9\).
So, the best description aligns with the last option:
The product exponent rule to add \(5\) and \(-4\) to get an exponent of \(1\) on the \(x\), while also considering the multiplication of the coefficients.
In summary, you should focus on the multiplication of coefficients and the product rule for adding exponents.
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