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

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.

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

Learn how to simplify the algebraic expression \frac{3a^2(x^3y)^4}{(6a \cdot x)^2y^3} step by step. This solution covers key algebra concepts, including exponent rules and simplifying rational expressions. Suitable for students in Grades 9-12.

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