Find the modulus of ` ((3+2i)^(2))/(4-3i)`

Answer

To find the modulus of \(\frac{(3 + 2i)^2}{4 - 3i}\), we can follow these steps:

### Step 1: Use the property of modulus
We know that the modulus of a quotient of two complex numbers can be expressed as:
\[ \left| \frac{Z_1}{Z_2} \right| = \frac{|Z_1|}{|Z_2|} \]
Thus, we can rewrite the expression:
\[ \left| \frac{(3 + 2i)^2}{4 - 3i} \right| = \frac{|(3 + 2i)^2|}{|4 - 3i|} \]

### Step 2: Calculate the modulus of \(3 + 2i\)
The modulus of a complex number \(a + bi\) is given by:
\[ |a + bi| = \sqrt{a^2 + b^2} \]
For \(3 + 2i\):
\[ |3 + 2i| = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \]

### Step 3: Calculate the modulus of \((3 + 2i)^2\)
Using the property of modulus:
\[ |(3 + 2i)^2| = |3 + 2i|^2 = (\sqrt{13})^2 = 13 \]

### Step 4: Calculate the modulus of \(4 - 3i\)
For \(4 - 3i\):
\[ |4 - 3i| = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 5: Combine the results
Now we can substitute the moduli back into our expression:
\[ \left| \frac{(3 + 2i)^2}{4 - 3i} \right| = \frac{|(3 + 2i)^2|}{|4 - 3i|} = \frac{13}{5} \]

### Final Answer
Thus, the modulus of \(\frac{(3 + 2i)^2}{4 - 3i}\) is:
\[ \frac{13}{5} \]