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 complex number \(\frac{(2 + 3i)^2}{2 + i}\), we will follow these steps:

### Step 1: Calculate \((2 + 3i)^2\)
To square the complex number \(2 + 3i\), we use the formula \((a + bi)^2 = a^2 + 2abi + (bi)^2\).

\[ (2 + 3i)^2 = 2^2 + 2(2)(3i) + (3i)^2 \]
\[ = 4 + 12i + 9i^2 \]
Since \(i^2 = -1\), we have:
\[ = 4 + 12i - 9 = -5 + 12i \]

### Step 2: Find the modulus of \(-5 + 12i\)
The modulus of a complex number \(a + bi\) is given by \(\sqrt{a^2 + b^2}\).

\[ |-5 + 12i| = \sqrt{(-5)^2 + (12)^2} \]
\[ = \sqrt{25 + 144} = \sqrt{169} = 13 \]

### Step 3: Calculate the modulus of \(2 + i\)
Now we find the modulus of the denominator \(2 + i\).

\[ |2 + i| = \sqrt{2^2 + 1^2} \]
\[ = \sqrt{4 + 1} = \sqrt{5} \]

### Step 4: Use the property of moduli
The modulus of a quotient of complex numbers is the quotient of their moduli. Thus,

\[ \left|\frac{(2 + 3i)^2}{2 + i}\right| = \frac{|(2 + 3i)^2|}{|2 + i|} = \frac{13}{\sqrt{5}} \]

### Final Answer
The modulus of \(\frac{(2 + 3i)^2}{2 + i}\) is \(\frac{13}{\sqrt{5}}\).

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