https://file.aiquickdraw.com/mathbot/file/1725721964455ypczrsfo.jpg

The expression given in the image is:

\[
\frac{(4x^2 y)(3x^2 y)}{(2x - 3y)^2}
\]

We are asked to simplify this expression and then evaluate it for \( x = 2 \) and \( y = 3 \).

### Step 1: Simplify the Expression

1. **Simplify the Numerator:**
   \[
   (4x^2 y)(3x^2 y) = 4 \cdot 3 \cdot x^2 \cdot x^2 \cdot y \cdot y = 12x^4y^2
   \]

2. **Simplify the Denominator:**
   \[
   (2x - 3y)^2 = (2x - 3y)(2x - 3y)
   \]

The simplified expression becomes:

\[
\frac{12x^4y^2}{(2x - 3y)^2}
\]

### Step 2: Evaluate for \( x = 2 \) and \( y = 3 \)

1. **Substitute \( x = 2 \) and \( y = 3 \) into the expression:**

\[
\frac{12(2)^4(3)^2}{(2(2) - 3(3))^2}
\]

2. **Calculate each part:**

- \( 2^4 = 16 \)
- \( 3^2 = 9 \)
- Numerator: \( 12 \cdot 16 \cdot 9 = 12 \cdot 144 = 1728 \)

- Denominator: \( (4 - 9)^2 = (-5)^2 = 25 \)

3. **Combine to find the final value:**

\[
\frac{1728}{25} = 69.12
\]

### Final Answer

The simplified expression is:

\[
\frac{12x^4y^2}{(2x - 3y)^2}
\]

And its value for \( x = 2 \) and \( y = 3 \) is:

\[
69.12
\]

Would you like more details or have any questions about this problem?

### Related Questions:

1. How do you simplify expressions involving variables and exponents?
2. What is the process to expand binomials like \( (2x - 3y)^2 \)?
3. Why is it important to simplify before substituting values?
4. How do you handle negative values in exponents or powers?
5. What techniques can be used to check the correctness of an algebraic simplification?

#### Tip:
Always double-check your substitution values to avoid errors, especially when dealing with negative signs!

Simplify and then evaluate the expression \( \frac{(4x^2 y)(3x^2 y)}{(2x - 3y)^2} \) for \( x = 2 \) and \( y = 3 \).

Simplify and evaluate the algebraic expression \( \frac{(4x^2 y)(3x^2 y)}{(2x - 3y)^2} \) for \( x = 2 \) and \( y = 3 \). This problem involves algebraic simplification, exponent rules, and substitution, suitable for students in Grades 9-10.

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