To multiply and simplify $(1 + 2i) \cdot (1 - 4i)$, we apply the distributive property (or FOIL method):

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

Step 1: Expand the expression

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

Step 2: Combine terms

Substituting back into the expression: $1 - 4i + 2i - 8i^2$

Step 3: Simplify using $i^2 = -1$

Since $i^2 = -1$, substitute $-1$ for $i^2$: $1 - 4i + 2i - 8(-1)$ $1 - 4i + 2i + 8$

Step 4: Combine real and imaginary terms

$(1 + 8) + (-4i + 2i)$ $9 - 2i$

Final Answer:

$\boxed{9 - 2i}$

Would you like a step-by-step explanation for any specific part? 😊

Related Questions:

1. How does $i^2 = -1$ affect complex number calculations?
2. What is the geometric interpretation of multiplying two complex numbers?
3. Can you demonstrate the FOIL method with another complex number example?
4. What are the rules for simplifying powers of $i$?
5. How does the distributive property work with imaginary numbers?

Tip:

Always simplify powers of $i$ by remembering the pattern: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, and then it repeats!