\frac{\left(2x^2\cdot y\right)^2\cdot 5x\cdot y}{\left(3x\cdot y^3\right)^3}

Let's simplify the given expression:

\[
\frac{\left(2x^2\cdot y\right)^2\cdot 5x\cdot y}{\left(3x\cdot y^3\right)^3}
\]

### Step 1: Simplify the numerator and the denominator separately.

#### Numerator:
\[
\left(2x^2 \cdot y\right)^2 \cdot 5x \cdot y
\]

Simplify \(\left(2x^2 \cdot y\right)^2\):
\[
\left(2x^2 \cdot y\right)^2 = (2^2) \cdot (x^2)^2 \cdot y^2 = 4x^4 y^2
\]

Now, multiply by \(5x \cdot y\):
\[
4x^4 y^2 \cdot 5x \cdot y = 20x^5 y^3
\]

#### Denominator:
\[
\left(3x \cdot y^3\right)^3
\]

Simplify \(\left(3x \cdot y^3\right)^3\):
\[
\left(3x \cdot y^3\right)^3 = (3^3) \cdot (x^3) \cdot (y^3)^3 = 27x^3 y^9
\]

### Step 2: Combine the simplified numerator and denominator.

\[
\frac{20x^5 y^3}{27x^3 y^9}
\]

### Step 3: Simplify the fraction.

Simplify the powers of \(x\) and \(y\):
\[
\frac{20x^5}{27x^3} = \frac{20x^{5-3}}{27} = \frac{20x^2}{27}
\]

\[
\frac{y^3}{y^9} = y^{3-9} = y^{-6} = \frac{1}{y^6}
\]

Combine these results:
\[
\frac{20x^2}{27y^6}
\]

### Final Simplified Expression:

\[
\frac{20x^2}{27y^6}
\]

Would you like more details or have any questions?

Here are some related questions to consider:

1. How do you simplify expressions with negative exponents?
2. What are the rules for multiplying exponents?
3. How does distributing exponents work inside parentheses?
4. How do you simplify complex fractions?
5. What are common mistakes to avoid when working with exponents?

**Tip:** Always simplify the powers of each variable separately to avoid errors in calculations.

This guide simplifies the algebraic expression \frac{(2x^2 \cdot y)^2 \cdot 5x \cdot y}{(3x \cdot y^3)^3}, covering key concepts like exponentiation, multiplication, and division of variables with exponents. Suitable for students in Grades 9-12.

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