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The given problem asks you to simplify the expression \((6 - 8i)^2\) into the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.

Let's break it down step by step:

### Step 1: Expand the square
You can expand \((6 - 8i)^2\) using the formula \((x - y)^2 = x^2 - 2xy + y^2\).
So, applying it to this expression:
\[
(6 - 8i)^2 = 6^2 - 2(6)(8i) + (8i)^2
\]

### Step 2: Simplify the terms
Now calculate each term:
- \(6^2 = 36\)
- \(-2(6)(8i) = -96i\)
- \((8i)^2 = 64i^2\)

Remember, \(i^2 = -1\), so:
- \(64i^2 = 64(-1) = -64\)

### Step 3: Combine the real and imaginary parts
Now put everything together:
\[
36 - 96i - 64 = (36 - 64) - 96i = -28 - 96i
\]

So the simplified expression is:
\[
-28 - 96i
\]

Thus, the answer in \(a + bi\) form is:
\[
\boxed{-28 - 96i}
\]

Would you like any further clarification or details?

Here are 5 related questions:
1. How would you simplify \((3 + 4i)^2\)?
2. What happens when you square a purely imaginary number like \(7i\)?
3. Can you generalize the formula for squaring a complex number \((a + bi)^2\)?
4. How do you multiply two different complex numbers?
5. How do you convert a complex number from rectangular form \(a + bi\) to polar form?

**Tip**: Always remember that \(i^2 = -1\) when working with complex numbers!

Simplify the expression (6 - 8i)^2 to a + bi form.

Learn how to simplify the complex number expression (6 - 8i)^2 into the form a + bi. This detailed step-by-step solution breaks down key algebraic concepts and uses the formula (x - y)^2 to simplify complex numbers.

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