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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 simplify the product

[tex]$$
(8-5i)^2,
$$[/tex]

follow these steps:

1. Write the square as a product of two factors:

[tex]$$
(8-5i)^2 = (8-5i)(8-5i).
$$[/tex]

2. Expand by applying the formula for the square of a binomial:

[tex]$$
(8-5i)^2 = 8^2 + 2\cdot8\cdot(-5i) + (-5i)^2.
$$[/tex]

3. Calculate each term:
- The square of the real part:

[tex]$$
8^2 = 64.
$$[/tex]

- Twice the product of the real and imaginary parts:

[tex]$$
2\cdot8\cdot(-5i) = -80i.
$$[/tex]

- The square of the imaginary part:

[tex]$$
(-5i)^2 = (-5)^2\cdot i^2 = 25\cdot(-1) = -25 \quad \text{(since } i^2 = -1\text{)}.
$$[/tex]

4. Combine all the computed terms:

[tex]$$
64 - 80i - 25.
$$[/tex]

5. Simplify by combining the real numbers:

[tex]$$
64 - 25 = 39,
$$[/tex]

so the expression becomes:

[tex]$$
39 - 80i.
$$[/tex]

Thus, the simplified product is

[tex]$$
\boxed{39-80i}.
$$[/tex]