Taking the value of the square root with positive real part only, the value of `sqrt(7+24i)+sqrt(-7-24i)`, is

Options

A. `1+7i`
B. `-1-7i`
C. `7-i`
D. `-7+i`

Explanation

To solve the expression \( \sqrt{7 + 24i} + \sqrt{-7 - 24i} \), we will follow a systematic approach to find the square roots of the complex numbers involved.

### Step 1: Calculate \( \sqrt{7 + 24i} \)

1. **Find the modulus**:
\[ |z| = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \]

2. **Use the formula for the square root of a complex number**:
The square root of a complex number \( z = a + bi \) can be expressed as:
\[ \sqrt{z} = \pm \left( \sqrt{\frac{|z| + a}{2}} + i \sqrt{\frac{|z| - a}{2}} \right) \]
Here, \( a = 7 \) and \( b = 24 \).

3. **Substituting values**:
\[ \sqrt{7 + 24i} = \sqrt{\frac{25 + 7}{2}} + i \sqrt{\frac{25 - 7}{2}} = \sqrt{\frac{32}{2}} + i \sqrt{\frac{18}{2}} = \sqrt{16} + i \sqrt{9} = 4 + 3i \]

### Step 2: Calculate \( \sqrt{-7 - 24i} \)

1. **Find the modulus**:
\[ |-7 - 24i| = \sqrt{(-7)^2 + (-24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \]

2. **Use the formula for the square root**:
Here, \( a = -7 \) and \( b = -24 \).
\[ \sqrt{-7 - 24i} = \pm \left( \sqrt{\frac{25 - 7}{2}} + i \sqrt{\frac{25 + 7}{2}} \right) \]

3. **Substituting values**:
\[ \sqrt{-7 - 24i} = \sqrt{\frac{25 - 7}{2}} + i \sqrt{\frac{25 + 7}{2}} = \sqrt{\frac{18}{2}} + i \sqrt{\frac{32}{2}} = 3i + 4 \]
Since we want the square root with a positive real part, we take:
\[ \sqrt{-7 - 24i} = 3 - 4i \]

### Step 3: Combine the results

Now we combine the two results:
\[ \sqrt{7 + 24i} + \sqrt{-7 - 24i} = (4 + 3i) + (3 - 4i) = 4 + 3i + 3 - 4i = 7 - i \]

### Final Answer
Thus, the value of \( \sqrt{7 + 24i} + \sqrt{-7 - 24i} \) is:
\[ \boxed{7 - i} \]