Find the modulus of the following complex numbers
`(3 +2i) (5-4i)`

Answer

To find the modulus of the complex number \( (3 + 2i)(5 - 4i) \), we can follow these steps:

### Step 1: Use the property of modulus
The modulus of the product of two complex numbers is equal to the product of their moduli. Therefore, we can write:
\[ |Z| = |(3 + 2i)(5 - 4i)| = |3 + 2i| \cdot |5 - 4i| \]

### Step 2: Calculate the modulus of \( 3 + 2i \)
Using the formula for the modulus of a complex number \( x + yi \), which is given by \( \sqrt{x^2 + y^2} \):
\[ |3 + 2i| = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \]

### Step 3: Calculate the modulus of \( 5 - 4i \)
Similarly, we calculate the modulus of \( 5 - 4i \):
\[ |5 - 4i| = \sqrt{5^2 + (-4)^2} = \sqrt{25 + 16} = \sqrt{41} \]

### Step 4: Combine the results
Now we can combine the results from Steps 2 and 3:
\[ |Z| = |3 + 2i| \cdot |5 - 4i| = \sqrt{13} \cdot \sqrt{41} \]

### Step 5: Simplify the expression
Using the property of square roots, we can simplify:
\[ |Z| = \sqrt{13 \cdot 41} = \sqrt{533} \]

### Final Answer
Thus, the modulus of the complex number \( (3 + 2i)(5 - 4i) \) is:
\[ \sqrt{533} \]

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