If `(4 + 3i)^(2) = 7 + 24i`, then a value of `(7 + sqrt(-576))^(1//2) - (7-sqrt(-576))^(1//2)` is :

Options

A. `-6i`
B. `-3i`
C. 2i
D. 6

Answer

To solve the problem, we need to evaluate the expression \( (7 + \sqrt{-576})^{1/2} - (7 - \sqrt{-576})^{1/2} \).

### Step-by-step Solution:

1. **Evaluate \(\sqrt{-576}\)**:
\[ \sqrt{-576} = \sqrt{-1 \cdot 576} = \sqrt{-1} \cdot \sqrt{576} = i \cdot 24 = 24i \]
**Hint**: Remember that \(\sqrt{-1} = i\) and \(\sqrt{576} = 24\).

2. **Substitute \(\sqrt{-576}\) into the expression**:
\[ (7 + \sqrt{-576})^{1/2} - (7 - \sqrt{-576})^{1/2} = (7 + 24i)^{1/2} - (7 - 24i)^{1/2} \]

3. **Recognize that \(7 + 24i\) can be related to \((4 + 3i)^2\)**:
From the problem statement, we know that:
\[ (4 + 3i)^2 = 4^2 + 2 \cdot 4 \cdot 3i + (3i)^2 = 16 + 24i - 9 = 7 + 24i \]
Thus, we can write:
\[ (7 + 24i)^{1/2} = \pm (4 + 3i) \]
**Hint**: Use the property of square roots that \((a + bi)^2 = a^2 - b^2 + 2abi\).

4. **Similarly, find \((7 - 24i)^{1/2}\)**:
We can find the square root of \(7 - 24i\) by recognizing that it is the conjugate of \(7 + 24i\):
\[ (7 - 24i)^{1/2} = \pm (4 - 3i) \]

5. **Substituting back into the expression**:
Now we substitute back:
\[ (7 + 24i)^{1/2} - (7 - 24i)^{1/2} = (4 + 3i) - (4 - 3i) \]

6. **Simplify the expression**:
\[ (4 + 3i) - (4 - 3i) = 4 + 3i - 4 + 3i = 6i \]

7. **Conclusion**:
The final value of the expression is:
\[ 6i \]

### Final Answer:
\[ 6i \]