To simplify the given expression:

$\frac{3a^2\left(x^3y\right)^4}{\left(6a \cdot x\right)^2 y^3}$

Let's break it down step by step.

Step 1: Expand the numerator and denominator.

• For the numerator: $3a^2 \left(x^3 y\right)^4 = 3a^2 \cdot x^{12} \cdot y^4$
• For the denominator: $\left(6a \cdot x\right)^2 y^3 = (6^2) \cdot a^2 \cdot x^2 \cdot y^3 = 36a^2x^2y^3$

Thus, the expression becomes: $\frac{3a^2 x^{12} y^4}{36 a^2 x^2 y^3}$

Step 2: Simplify the coefficients, variables, and powers.

• The coefficient: $\frac{3}{36} = \frac{1}{12}$
• The $a^2$ terms cancel out.
• For the $x$ terms: $\frac{x^{12}}{x^2} = x^{12-2} = x^{10}$
• For the $y$ terms: $\frac{y^4}{y^3} = y^{4-3} = y$

Thus, the simplified expression is: $\frac{x^{10}y}{12}$

Final answer:

$\frac{x^{10}y}{12}$

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. Simplify: $\frac{2x^4y^2}{4x^2y}$
2. Expand and simplify: $\frac{a^3(b^2c)^3}{(abc)^2}$
3. Simplify: $\frac{5m^2n^3(mn^2)^2}{10m^4n^5}$
4. Factor: $x^4 + 4x^2 + 4$
5. Simplify: $\frac{9a^5b^2}{3a^3b}$

Tip: When simplifying, always look to cancel common terms in both the numerator and denominator.