Simplify a¹² ÷ a⁹ × a³ ÷ a⁴: Exponent Manipulation Practice

We'll begin by applying the multiplication law between fractions, multiplying numerator by numerator and denominator by denominator:

$\frac{x}{y}\cdot\frac{w}{z}=\frac{x\cdot w}{y\cdot z}$

Let's return to the problem and apply the above law:

$\frac{a^{12}}{a^9}\cdot\frac{a^3}{a^4}=\frac{a^{12}\cdot a^3}{a^{^9}\cdot a^4}$

From here on we will no longer indicate the multiplication sign, instead we will place the terms next to each other.

Note that in both the numerator and denominator, multiplication is performed between terms with identical bases, therefore we'll apply the power law for multiplication between terms with the same base:

$b^m\cdot b^n=b^{m+n}$

Note that this law can only be used to calculate multiplication between terms with identical bases.

Let's return to the problem and calculate separately the results of the multiplication in the numerator and denominator:

$\frac{a^{12}a^3}{a^{^9}a^4}=\frac{a^{12+3}}{a^{9+4}}=\frac{a^{15}}{a^{13}}$

In the last step we calculated the sum of the exponents.

Now we need to perform division (fraction=division operation between numerator and denominator) between terms with identical bases, therefore we'll apply the power law for division between terms with the same base:

$\frac{b^m}{b^n}=b^{m-n}$

Note that this law can only be used to calculate division between terms with identical bases.

Let's return to the problem and apply the above law:

$\frac{a^{15}}{a^{13}}=a^{15-13}=a^2$

In the last step we calculated the result of the subtraction operation in the exponent.

We cannot simplify the expression further. Therefore the correct answer is D.