We are tasked with simplifying the expression $(2 + 3i) \cdot (1 - 4i)$. Let's go step by step.

Step 1: Use the distributive property (FOIL method)

We distribute the terms: $(2 + 3i)(1 - 4i) = 2(1) + 2(-4i) + 3i(1) + 3i(-4i)$ Now simplify each term:

• $2 \cdot 1 = 2$
• $2 \cdot (-4i) = -8i$
• $3i \cdot 1 = 3i$
• $3i \cdot (-4i) = -12i^2$

Step 2: Simplify using $i^2 = -1$

Recall that $i^2 = -1$, so: $-12i^2 = -12(-1) = 12$

Step 3: Combine all terms

Now substitute back into the expression: $2 - 8i + 3i + 12$ Combine the real terms and the imaginary terms: $(2 + 12) + (-8i + 3i) = 14 - 5i$

Final Answer:

$(2 + 3i) \cdot (1 - 4i) = 14 - 5i$

Would you like more details on this process or clarification on any step?

Here are 5 related questions you could explore:

1. What happens when you multiply two complex numbers geometrically?
2. How do you find the conjugate of a complex number?
3. Can you simplify $(3 + 2i)(2 + 5i)$?
4. What is the modulus of the complex number $14 - 5i$?
5. How do you divide two complex numbers?

Tip: When multiplying complex numbers, always remember that $i^2 = -1$, which simplifies the terms with imaginary units.