To simplify the given expression:

$\frac{3x^{14}y^{11}}{18x^2}$

we can follow these steps:

Step 1: Simplify the Coefficients

The coefficient in the numerator is $3$ and in the denominator is $18$. Simplifying $\frac{3}{18}$ gives us $\frac{1}{6}$.

So the expression becomes: $\frac{x^{14}y^{11}}{6x^2}$

Step 2: Simplify the Variables

For the $x$-terms, we have $x^{14}$ in the numerator and $x^2$ in the denominator. Using the property of exponents $\frac{x^a}{x^b} = x^{a - b}$, we get: $x^{14 - 2} = x^{12}$

This gives us: $\frac{x^{12}y^{11}}{6}$

Final Expression

The simplified expression is: $\frac{x^{12}y^{11}}{6}$

Answer

The correct choice is D: $\frac{x^{12}y^{11}}{6}$.

Would you like further explanation on any part? Here are some related questions for practice:

1. How would you simplify $\frac{5x^{10}y^{8}}{15x^3y^2}$?
2. What is the result of $\frac{6x^{15}y^5}{9x^4y^2}$?
3. How do you simplify an expression where both the numerator and denominator have terms with negative exponents?
4. How would you approach simplifying $\frac{4x^{20}y^{18}}{8x^5y^8}$?
5. Can you explain the rule $\frac{x^a}{x^b} = x^{a-b}$ and why it works?

Tip: Always simplify coefficients first, then handle the variable terms separately for a clear and systematic approach.