simplify the expression to a + bi form: (-2-6i)^2

To simplify the expression \((-2 - 6i)^2\) into the form \(a + bi\), we will expand the square of the binomial:

1. **Expand the square using the formula \((a + b)^2 = a^2 + 2ab + b^2\):**
   \[
   (-2 - 6i)^2 = (-2)^2 + 2(-2)(-6i) + (-6i)^2
   \]

2. **Calculate each term:**
   - \((-2)^2 = 4\)
   - \(2(-2)(-6i) = 24i\)
   - \((-6i)^2 = 36i^2\)

   Since \(i^2 = -1\), substitute \(-1\) for \(i^2\):
   \[
   36i^2 = 36(-1) = -36
   \]

3. **Combine the real and imaginary parts:**
   \[
   = 4 + 24i - 36
   \]
   \[
   = (4 - 36) + 24i
   \]
   \[
   = -32 + 24i
   \]

So, the simplified form of \((-2 - 6i)^2\) is:

\[
-32 + 24i
\]

Would you like more details or have any questions? Here are some related questions:

1. How do you find the magnitude of a complex number?
2. What happens when you square a complex number?
3. How do you multiply two complex numbers in polar form?
4. How can complex conjugates be used to simplify expressions?
5. What is the significance of the imaginary unit \(i\) in mathematics?

**Tip:** When squaring a complex number, be careful with the signs, especially when dealing with \(i^2 = -1\).

Learn how to simplify the complex expression (-2 - 6i)^2 into the standard form a + bi with detailed steps. This problem involves complex numbers and binomial expansion, suitable for advanced high school students and beyond.

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