The modulus of ` ((2+3i)^(2))/(2+i)` is

Options

A. `sqrt13/5`
B. `sqrt147/5`
C. `13/sqrt5`
D. `sqrt185/5`

Answer

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

### Step 1: Use the properties of modulus
We will use the properties of modulus:
1. \(|z^2| = |z|^2\)
2. \(\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}\)

Using these properties, we can rewrite the modulus of the expression:
\[ \left|\frac{(2 + 3i)^2}{2 + i}\right| = \frac{|(2 + 3i)^2|}{|2 + i|} \]

### Step 2: Calculate \(|2 + 3i|\)
To find \(|2 + 3i|\), we use the formula for the modulus of a complex number:
\[ |a + bi| = \sqrt{a^2 + b^2} \]
Here, \(a = 2\) and \(b = 3\):
\[ |2 + 3i| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \]

### Step 3: Calculate \(|(2 + 3i)^2|\)
Now we can find \(|(2 + 3i)^2|\):
\[ |(2 + 3i)^2| = |2 + 3i|^2 = (\sqrt{13})^2 = 13 \]

### Step 4: Calculate \(|2 + i|\)
Next, we calculate \(|2 + i|\):
\[ |2 + i| = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \]

### Step 5: Substitute back into the modulus expression
Now we substitute back into our modulus expression:
\[ \left|\frac{(2 + 3i)^2}{2 + i}\right| = \frac{|(2 + 3i)^2|}{|2 + i|} = \frac{13}{\sqrt{5}} \]

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

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