Find `(3m+1/(5))^(2)=` _____

Options

A. `3m^(2)+(6m)/(5)+1/(25)`
B. `9m^(2)+(3m)/(5)-1/(25)`
C. `9m^(2)+(6m)/(5)+1/(25)`
D. `3m^(2)+(3m)/(5)+1/(25)`

Answer

To find the value of \((3m + \frac{1}{5})^2\), we will use the algebraic identity for the square of a binomial. The identity states:

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

### Step-by-Step Solution:

1. **Identify \(a\) and \(b\)**:
- Here, \(a = 3m\) and \(b = \frac{1}{5}\).

2. **Calculate \(a^2\)**:
- \(a^2 = (3m)^2 = 9m^2\).

3. **Calculate \(b^2\)**:
- \(b^2 = \left(\frac{1}{5}\right)^2 = \frac{1}{25}\).

4. **Calculate \(2ab\)**:
- \(2ab = 2 \times (3m) \times \left(\frac{1}{5}\right) = \frac{6m}{5}\).

5. **Combine all parts**:
- Now, we combine these results using the identity:
\[ (3m + \frac{1}{5})^2 = a^2 + 2ab + b^2 = 9m^2 + \frac{6m}{5} + \frac{1}{25} \]

6. **Final Expression**:
- Therefore, the final expression is:
\[ 9m^2 + \frac{6m}{5} + \frac{1}{25} \]

### Final Answer:
\[ (3m + \frac{1}{5})^2 = 9m^2 + \frac{6m}{5} + \frac{1}{25} \]