Power of a Quotient Rule for Exponents: Number of terms

First, we'll enter the same fraction using the multiplication law between fractions, by 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, but use the conventional writing method where placing terms next to each other means multiplication.

Now we'll notice that both in the numerator and denominator, multiplication is performed between terms with identical bases, therefore we'll use 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 multiplication in the numerator and denominator:

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

where in the last step we calculated the sum of the exponents.

Now, we'll notice that we need to perform division (fraction=division operation between numerator and denominator) between terms with identical bases, therefore we'll use 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:

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

where in the last step we calculated the result of subtraction in the exponent.

We got the most simplified expression possible and therefore we're done,

therefore the correct answer is D.

First, let's do some cosmetic work and arrange the problem into a familiar form. We'll write the expression in an organized way using fractions, remembering that division is actually multiplication by the reciprocal, and therefore instead of dividing by a fraction we can always multiply by its reciprocal. We'll also remember that to get the reciprocal of a simple fraction we just flip between the numerator and denominator. Mathematically, instead of writing:$:\frac{x}{y}$We can always write:$\cdot\frac{y}{x}$

Let's apply this to the problem:

$\frac{b^{10}}{b^2}:\frac{b^9}{b^5}=\frac{b^{10}}{b^2}\cdot\frac{b^5}{b^9}$F

rom here the solution becomes clear, we'll continue treating each of the fractions we got between which multiplication is performed,

Notice that in both fractions there are terms in the numerator and denominator with identical bases, so we'll use the division law for terms with identical bases to simplify the expression:

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

Let's apply this law to each fraction separately:

$\frac{b^{10}}{b^2}\cdot\frac{b^5}{b^9}=b^{10-2}\cdot b^{5-9}=b^8\cdot b^{-4}$

Where in the second stage we calculated the result of the subtraction operation in the exponents for each term separately,

Next, we'll notice that we want to calculate multiplication between two terms with the same base, so we'll use the power law for multiplication between terms with identical bases:

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

Let's apply this law to the problem:

$b^8\cdot b^{-4}=b^{8+(-4)}=b^{8-4}=b^4$

We got the most simplified expression, so we're done.

Therefore the correct answer is C.