Simplify y³/y⁶ × y⁴/y⁻² × y¹²/y⁷: Complete Exponent Operation

We need to calculate division (fraction=division operation between numerator and denominator) between terms with identical bases, therefore we will use the law of exponents for division between terms with identical 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.

In this problem, there is also a term with a negative exponent, but this does not pose an issue regarding the use of the aforementioned law of exponents. In fact, this law of exponents is valid in all cases for numerical terms with different exponents, including negative exponents, rational number exponents, and even irrational number exponents, etc.

Let's return to the problem and apply the aforementioned law of exponents for each fraction separately:

$\frac{y^3}{y^6}\cdot\frac{y^4}{y^{-2}}\cdot\frac{y^{12}}{y^7}=y^{3-6}\cdot y^{4-(-2)}\cdot y^{12-7}=y^{-3}\cdot y^6\cdot y^5$

When in the second stage we applied the aforementioned law of exponents for the second fraction (from left to right) carefully, this is because the term in the denominator of this fraction has a negative exponent and according to the aforementioned law of exponents, we need to subtract between the exponent of the numerator and the exponent of the denominator, which in this case gave us subtraction of a negative number from another number, an operation we performed carefully.

From here on we will no longer indicate the multiplication sign, but use the conventional writing form where placing terms next to each other means multiplication.

Let's return to the problem and note that we need to perform multiplication between terms with identical bases, therefore we will use the law of exponents for multiplication between terms with identical base:

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

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

Let's apply this law in the problem:

$y^{-3}y^6y^5=y^{-3+6+5}=y^8$

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

Therefore the correct answer is B.