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Multiply and simplify the product of the complex numbers:

[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], follow these steps:

1. Write out the expression for squaring:
[tex]\((8 - 5i)^2 = (8 - 5i)(8 - 5i)\)[/tex]

2. Use the distributive property (FOIL method):
Multiply each part of the first binomial by each part of the second binomial:
[tex]\[
(8 - 5i)(8 - 5i) = 8 \times 8 + 8 \times (-5i) + (-5i) \times 8 + (-5i) \times (-5i)
\][/tex]

3. Calculate each term:
- [tex]\(8 \times 8 = 64\)[/tex]
- [tex]\(8 \times (-5i) = -40i\)[/tex]
- [tex]\((-5i) \times 8 = -40i\)[/tex]
- [tex]\((-5i) \times (-5i) = 25i^2\)[/tex]

4. Remember that [tex]\(i^2 = -1\)[/tex]:
[tex]\((-5i) \times (-5i) = 25(-1) = -25\)[/tex]

5. Combine like terms:
- The real part: [tex]\(64 - 25 = 39\)[/tex]
- The imaginary part: [tex]\(-40i - 40i = -80i\)[/tex]

6. Combine the results:
[tex]\((8 - 5i)^2 = 39 - 80i\)[/tex]

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

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