Find the square root of the following complex number
`((2+3i)/(5-4i) + (2-3i)/(5+4i))`

Answer

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

### Step 1: Combine the fractions
We start by finding a common denominator for the two fractions. The common denominator is \((5 - 4i)(5 + 4i)\).

\[ S = \frac{(2 + 3i)(5 + 4i) + (2 - 3i)(5 - 4i)}{(5 - 4i)(5 + 4i)} \]

### Step 2: Expand the numerators
Now we will expand the numerators of both fractions.

1. For \((2 + 3i)(5 + 4i)\):
\[ = 2 \cdot 5 + 2 \cdot 4i + 3i \cdot 5 + 3i \cdot 4i = 10 + 8i + 15i + 12i^2 \]
Since \(i^2 = -1\), we have:
\[ = 10 + 23i - 12 = -2 + 23i \]

2. For \((2 - 3i)(5 - 4i)\):
\[ = 2 \cdot 5 - 2 \cdot 4i - 3i \cdot 5 + 3i \cdot 4i = 10 - 8i - 15i + 12i^2 \]
Again, using \(i^2 = -1\):
\[ = 10 - 23i - 12 = -2 - 23i \]

### Step 3: Combine the expanded numerators
Now we combine the results from the two expansions:
\[ S = \frac{(-2 + 23i) + (-2 - 23i)}{(5 - 4i)(5 + 4i)} \]
The imaginary parts cancel out:
\[ = \frac{-4}{(5 - 4i)(5 + 4i)} \]

### Step 4: Simplify the denominator
Now we simplify the denominator:
\[ (5 - 4i)(5 + 4i) = 5^2 - (4i)^2 = 25 - (-16) = 25 + 16 = 41 \]

### Step 5: Final expression for S
Thus, we have:
\[ S = \frac{-4}{41} \]

### Step 6: Find the square root of S
Now we need to find the square root of \(S\):
\[ \sqrt{S} = \sqrt{\frac{-4}{41}} = \frac{\sqrt{-4}}{\sqrt{41}} = \frac{\sqrt{4} \cdot \sqrt{-1}}{\sqrt{41}} = \frac{2i}{\sqrt{41}} \]

### Final Result
Thus, the square root of the given complex number is:
\[ \sqrt{S} = \pm \frac{2i}{\sqrt{41}} \]