Simplify: `(3x + 2/(4y))^(2) -(3x -2/(4y))^(2)`

Options

A. `(3x)/y`
B. `(3x)/(2y)`
C. `(6x)/y`
D. `(9x)/(4y)`

Answer

To simplify the expression \((3x + \frac{2}{4y})^2 - (3x - \frac{2}{4y})^2\), we can use the difference of squares formula, which states that \(a^2 - b^2 = (a - b)(a + b)\).

### Step-by-Step Solution:

1. **Identify \(a\) and \(b\)**:
Let \(a = (3x + \frac{2}{4y})\) and \(b = (3x - \frac{2}{4y})\).

2. **Apply the difference of squares formula**:
\[ (3x + \frac{2}{4y})^2 - (3x - \frac{2}{4y})^2 = (a - b)(a + b) \]

3. **Calculate \(a - b\)**:
\[ a - b = (3x + \frac{2}{4y}) - (3x - \frac{2}{4y}) = \frac{2}{4y} + \frac{2}{4y} = \frac{4}{4y} = \frac{1}{y} \]

4. **Calculate \(a + b\)**:
\[ a + b = (3x + \frac{2}{4y}) + (3x - \frac{2}{4y}) = 3x + 3x = 6x \]

5. **Combine the results**:
\[ (3x + \frac{2}{4y})^2 - (3x - \frac{2}{4y})^2 = (a - b)(a + b) = \left(\frac{1}{y}\right)(6x) = \frac{6x}{y} \]

### Final Answer:
\[ \frac{6x}{y} \]