Questions: Multiply the complex numbers (-7+11i)(2-3i).

Transcript text: 1. Multiply the complex numbers $(-7+11 i)(2-3 i)$.

Solution

Solution Steps

To multiply two complex numbers, we use the distributive property (also known as the FOIL method for binomials). This involves multiplying each part of the first complex number by each part of the second complex number and then combining like terms. Specifically, we multiply the real parts together, the real part of the first by the imaginary part of the second, the imaginary part of the first by the real part of the second, and the imaginary parts together. Finally, we combine the real and imaginary results.

Step 1: Define the Complex Numbers

We start with the complex numbers $ z_1 = -7 + 11i $ and $ z_2 = 2 - 3i $.

Step 2: Apply the Distributive Property

To multiply $ z_1 $ and $ z_2 $, we use the distributive property: \[ (-7 + 11i)(2 - 3i) = (-7 \cdot 2) + (-7 \cdot -3i) + (11i \cdot 2) + (11i \cdot -3i) \]

Step 3: Perform the Multiplications

Calculate each term: \[ -7 \cdot 2 = -14 \] \[ -7 \cdot -3i = 21i \] \[ 11i \cdot 2 = 22i \] \[ 11i \cdot -3i = -33i^2 \]

Step 4: Simplify Using $ i^2 = -1 $

Since $ i^2 = -1 $: \[ -33i^2 = -33(-1) = 33 \]

Step 5: Combine Like Terms

Combine the real and imaginary parts: \[ -14 + 33 + (21i + 22i) = 19 + 43i \]