To solve \((3 + 8i)^2\), we'll use the formula for squaring a binomial, which is \((a + b)^2 = a^2 + 2ab + b^2\).

Given:
\[
(3 + 8i)^2
\]

Let's break it down:

1. **Square the real part**:
   \[
   3^2 = 9
   \]

2. **Square the imaginary part**:
   \[
   (8i)^2 = 64i^2
   \]
   Since \(i^2 = -1\), this becomes:
   \[
   64i^2 = 64(-1) = -64
   \]

3. **Multiply the real and imaginary parts and double the product**:
   \[
   2 \times 3 \times 8i = 48i
   \]

Now, sum all these components together:
\[
9 - 64 + 48i = -55 + 48i
\]

So, \((3 + 8i)^2 = -55 + 48i\).

Would you like any additional details or explanations?

Here are some related questions:
1. What is \((4 - 7i)^2\)?
2. How do you multiply two different complex numbers, like \((3 + 8i)(2 - 5i)\)?
3. What is the geometric interpretation of multiplying a complex number by itself?
4. How can you use the distributive property to multiply complex numbers?
5. What is the significance of the imaginary unit \(i\) in complex numbers?

**Tip:** When multiplying complex numbers, remember to apply the distributive property and the identity \(i^2 = -1\) carefully to simplify the expression.

Learn how to solve the complex number problem (3 + 8i)^2 step-by-step. This problem involves squaring a binomial with complex numbers, demonstrating the application of the squaring formula for binomials. Suitable for students in Grades 11-12.

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