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------------------------------------------------ Multiply and simplify the product: [tex](8 - 5i)^2[/tex]

Select the product:

A. 39
B. 89
C. 39 - 80i
D. 89 - 80i

Answer :

To multiply and simplify [tex]\((8 - 5i)^2\)[/tex], we can follow these steps:

1. Understand the expression: We need to find the square of the complex number [tex]\(8 - 5i\)[/tex].

2. Use the formula for squaring a binomial:
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]

Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].

3. Apply the formula:
[tex]\[
(8 - 5i)^2 = 8^2 - 2 \cdot 8 \cdot 5i + (5i)^2
\][/tex]

4. Calculate each part:
- [tex]\(8^2 = 64\)[/tex]
- [tex]\(2 \cdot 8 \cdot 5i = 80i\)[/tex]
- [tex]\((5i)^2 = 25i^2\)[/tex]

5. Simplify using [tex]\(i^2 = -1\)[/tex]:
- [tex]\(25i^2 = 25 \times -1 = -25\)[/tex]

6. Combine all the parts:
[tex]\[
64 - 25 - 80i = 39 - 80i
\][/tex]

So, the product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex].

Therefore, the selected product is [tex]\(\boxed{39 - 80i}\)[/tex].