Find the modulus of the following
`(3-4i) (3 + 4i)`

Answer

To find the modulus of the complex number \( (3 - 4i)(3 + 4i) \), we can follow these steps:

### Step 1: Identify the complex numbers
Let \( z_1 = 3 - 4i \) and \( z_2 = 3 + 4i \).

### Step 2: Use the property of modulus
The modulus of the product of two complex numbers is equal to the product of their moduli:
\[ |z_1 z_2| = |z_1| \cdot |z_2| \]

### Step 3: Calculate the modulus of \( z_1 \)
The modulus of a complex number \( z = x + yi \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]
For \( z_1 = 3 - 4i \):
- \( x = 3 \)
- \( y = -4 \)

Calculating the modulus:
\[ |z_1| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 4: Calculate the modulus of \( z_2 \)
For \( z_2 = 3 + 4i \):
- \( x = 3 \)
- \( y = 4 \)

Calculating the modulus:
\[ |z_2| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 5: Calculate the modulus of the product
Now, we can find the modulus of the product:
\[ |z_1 z_2| = |z_1| \cdot |z_2| = 5 \cdot 5 = 25 \]

### Final Answer
Thus, the modulus of \( (3 - 4i)(3 + 4i) \) is \( 25 \).

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