Find the square roots of :
`((2+3i)/(5-4i)+(2-3i)/(5+4i))`.

Answer

To find the square roots of the expression \(\frac{(2+3i)}{(5-4i)} + \frac{(2-3i)}{(5+4i)}\), we will follow these steps:

### Step 1: Rationalize the Denominators
We need to rationalize both fractions.

1. For the first term \(\frac{(2+3i)}{(5-4i)}\):
\[ \frac{(2+3i)(5+4i)}{(5-4i)(5+4i)} \]
The denominator simplifies to:
\[ (5-4i)(5+4i) = 25 + 16 = 41 \]
The numerator simplifies to:
\[ (2+3i)(5+4i) = 10 + 8i + 15i + 12i^2 = 10 + 23i - 12 = -2 + 23i \]
Thus, the first term becomes:
\[ \frac{-2 + 23i}{41} \]

2. For the second term \(\frac{(2-3i)}{(5+4i)}\):
\[ \frac{(2-3i)(5-4i)}{(5+4i)(5-4i)} \]
The denominator, as calculated before, is also \(41\).
The numerator simplifies to:
\[ (2-3i)(5-4i) = 10 - 8i - 15i + 12i^2 = 10 - 23i - 12 = -2 - 23i \]
Thus, the second term becomes:
\[ \frac{-2 - 23i}{41} \]

### Step 2: Combine the Two Terms
Now we can combine the two terms:
\[ \frac{-2 + 23i}{41} + \frac{-2 - 23i}{41} = \frac{(-2 - 2) + (23i - 23i)}{41} = \frac{-4}{41} \]

### Step 3: Find the Square Root
Now we need to find the square root of \(-\frac{4}{41}\):
\[ \sqrt{-\frac{4}{41}} = \sqrt{-1} \cdot \sqrt{\frac{4}{41}} = i \cdot \frac{2}{\sqrt{41}} = \frac{2i}{\sqrt{41}} \]
Thus, the square roots are:
\[ \pm \frac{2i}{\sqrt{41}} \]

### Final Answer
The square roots of the given expression are:
\[ \pm \frac{2i}{\sqrt{41}} \]