Find the following products using the appropriate identity:
`((x)/(3)+(1)/(2))((x)/(3)-(1)/(2))`

Answer

To find the product \(\left(\frac{x}{3} + \frac{1}{2}\right)\left(\frac{x}{3} - \frac{1}{2}\right)\) using the appropriate identity, we can use the difference of squares identity, which states:

\[ (a + b)(a - b) = a^2 - b^2 \]

### Step-by-Step Solution:

1. **Identify \(a\) and \(b\)**:
- Here, \(a = \frac{x}{3}\) and \(b = \frac{1}{2}\).

2. **Apply the identity**:
- According to the identity, we can rewrite the product as:
\[ \left(\frac{x}{3} + \frac{1}{2}\right)\left(\frac{x}{3} - \frac{1}{2}\right) = \left(\frac{x}{3}\right)^2 - \left(\frac{1}{2}\right)^2 \]

3. **Calculate \(a^2\)**:
- Calculate \(\left(\frac{x}{3}\right)^2\):
\[ \left(\frac{x}{3}\right)^2 = \frac{x^2}{9} \]

4. **Calculate \(b^2\)**:
- Calculate \(\left(\frac{1}{2}\right)^2\):
\[ \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

5. **Combine the results**:
- Substitute \(a^2\) and \(b^2\) back into the identity:
\[ \frac{x^2}{9} - \frac{1}{4} \]

6. **Find a common denominator**:
- The common denominator for \(9\) and \(4\) is \(36\).
- Rewrite each term:
\[ \frac{x^2}{9} = \frac{4x^2}{36} \quad \text{and} \quad \frac{1}{4} = \frac{9}{36} \]

7. **Subtract the fractions**:
- Now, perform the subtraction:
\[ \frac{4x^2}{36} - \frac{9}{36} = \frac{4x^2 - 9}{36} \]

### Final Answer:
\[ \left(\frac{x}{3} + \frac{1}{2}\right)\left(\frac{x}{3} - \frac{1}{2}\right) = \frac{4x^2 - 9}{36} \]