To solve the expression \((2 - 2i)(8 + i)\), we'll use the distributive property (also known as the FOIL method for binomials):

\[
(2 - 2i)(8 + i) = 2(8) + 2(i) - 2i(8) - 2i(i)
\]

Now, let's calculate each term:

1. \(2 \times 8 = 16\)
2. \(2 \times i = 2i\)
3. \(-2i \times 8 = -16i\)
4. \(-2i \times i = -2i^2\)

Since \(i^2 = -1\), the last term becomes:

\[
-2i^2 = -2(-1) = 2
\]

Now, combine all the terms:

\[
16 + 2i - 16i + 2
\]

Combine the real and imaginary parts:

\[
(16 + 2) + (2i - 16i) = 18 - 14i
\]

So, the result is:

\[
\boxed{18 - 14i}
\]

Would you like further details or have any questions?

Here are 5 related questions you might find interesting:

1. What happens when you multiply two purely imaginary numbers?
2. How do you find the magnitude of a complex number?
3. What is the geometric interpretation of multiplying two complex numbers?
4. How do you add and subtract complex numbers?
5. How does multiplying by the complex conjugate affect a complex number?

**Tip:** When multiplying complex numbers, remember that \(i^2 = -1\) — this is key to simplifying your expressions.

Explore how to solve the expression (2 - 2i)(8 + i) step-by-step with detailed calculations and explanations. Learn about complex numbers, the distributive property, and the imaginary unit (i). Suitable for high school students studying algebra and complex numbers.

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