Simiplify : `(sqrt((5+12i)) + sqrt((5-12i)))/(sqrt((5+12i))-sqrt((5-12i)))`

Answer

To simplify the expression \((\sqrt{5 + 12i} + \sqrt{5 - 12i}) / (\sqrt{5 + 12i} - \sqrt{5 - 12i})\), we can follow these steps:

### Step 1: Identify the structure
Notice that the expression is in the form of \((a + b) / (a - b)\) where \(a = \sqrt{5 + 12i}\) and \(b = \sqrt{5 - 12i}\).

### Step 2: Multiply by the conjugate
To simplify, we multiply the numerator and the denominator by the conjugate of the denominator, which is \((\sqrt{5 + 12i} + \sqrt{5 - 12i})\):
\[ \frac{(\sqrt{5 + 12i} + \sqrt{5 - 12i})(\sqrt{5 + 12i} + \sqrt{5 - 12i})}{(\sqrt{5 + 12i} - \sqrt{5 - 12i})(\sqrt{5 + 12i} + \sqrt{5 - 12i})} \]

### Step 3: Apply the difference of squares
The denominator simplifies using the difference of squares:
\[ (\sqrt{5 + 12i})^2 - (\sqrt{5 - 12i})^2 = (5 + 12i) - (5 - 12i) = 24i \]

### Step 4: Expand the numerator
Now, expand the numerator:
\[ (\sqrt{5 + 12i} + \sqrt{5 - 12i})^2 = (\sqrt{5 + 12i})^2 + 2\sqrt{(5 + 12i)(5 - 12i)} + (\sqrt{5 - 12i})^2 \]
This becomes:
\[ (5 + 12i) + (5 - 12i) + 2\sqrt{(5 + 12i)(5 - 12i)} = 10 + 2\sqrt{(5 + 12i)(5 - 12i)} \]

### Step 5: Calculate the product under the square root
Next, calculate \((5 + 12i)(5 - 12i)\):
\[ (5 + 12i)(5 - 12i) = 5^2 - (12i)^2 = 25 - (-144) = 169 \]
Thus, \(\sqrt{(5 + 12i)(5 - 12i)} = \sqrt{169} = 13\).

### Step 6: Substitute back into the expression
Now substitute back into the numerator:
\[ 10 + 2 \cdot 13 = 10 + 26 = 36 \]

### Step 7: Final expression
Now, we can write the entire expression:
\[ \frac{36}{24i} \]
This simplifies to:
\[ \frac{36}{24} \cdot \frac{1}{i} = \frac{3}{2} \cdot \frac{1}{i} \]

### Step 8: Rationalize the denominator
To rationalize the denominator, multiply the numerator and denominator by \(i\):
\[ \frac{3i}{2i^2} = \frac{3i}{-2} = -\frac{3i}{2} \]

### Final Answer
Thus, the simplified form of the expression is:
\[ -\frac{3i}{2} \]