Simplify the Expression: 12b⁴ ÷ 4b⁻⁵ Using Power Rules

Let's consider that the numerator and the denominator of the fraction have terms with identical bases, therefore we will use the law of exponents for the division of terms with identical bases:

$\frac{c^m}{c^n}=c^{m-n}$We apply it to the problem:

$\frac{12b^4}{4b^{-5}}=3\cdot b^{4-(-5)}=3\cdot b^{4+5}=3b^9$When in the first step we simplify the numerical part of the fraction. This operation is intuitive as well as correct since it is possible to write down in advance the said fraction as a product of fractions and reduce:

$\frac{12b^4}{4b^{-5}}=\frac{12}{4}\cdot\frac{b^4}{b^{-5}}=3\cdot b^{4-(-5)}=\ldots$We return once again to the problem. The simplified expression obtained is as follows:

$3b^9$

$$Therefore, the correct answer is option D.