Find the modulus of the following using the property of modulus
`(3+4i) (8-6i)`

Answer

To find the modulus of the complex number expression \((3 + 4i)(8 - 6i)\), we can use the property of modulus of complex numbers. Here’s the step-by-step solution:

### Step 1: Identify the Complex Numbers
Let:
- \( z_1 = 3 + 4i \)
- \( z_2 = 8 - 6i \)

### Step 2: Find the Modulus of \( z_1 \)
The modulus of a complex number \( z = x + iy \) 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 3: Find the Modulus of \( z_2 \)
Now, for \( z_2 = 8 - 6i \):
- \( x = 8 \)
- \( y = -6 \)

Calculating the modulus:
\[ |z_2| = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]

### Step 4: Use the Property of Modulus
The property of modulus states that:
\[ |z_1 z_2| = |z_1| \cdot |z_2| \]
Thus, we can find the modulus of the product:
\[ |z_1 z_2| = |z_1| \cdot |z_2| = 5 \cdot 10 = 50 \]

### Final Answer
The modulus of the expression \((3 + 4i)(8 - 6i)\) is:
\[ \boxed{50} \]

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