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

Select the product:

A. 39
B. 89
C. [tex]39 - 80i[/tex]
D. [tex]89 - 80i[/tex]

Answer :

To simplify the expression

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

we can start by using the formula for the square of a binomial:

[tex]$$
(a-b)^2 = a^2 - 2ab + b^2.
$$[/tex]

In our case, [tex]$a = 8$[/tex] and [tex]$b = 5i$[/tex]. Now, let's compute each term step by step.

1. Calculate [tex]$a^2$[/tex]:
[tex]$$
8^2 = 64.
$$[/tex]

2. Calculate [tex]$-2ab$[/tex]:
[tex]$$
-2 \cdot 8 \cdot 5i = -80i.
$$[/tex]

3. Calculate [tex]$b^2$[/tex]:
[tex]$$
(5i)^2 = 25i^2.
$$[/tex]
Remember, [tex]$i^2 = -1$[/tex], so:
[tex]$$
25i^2 = 25 \times (-1) = -25.
$$[/tex]

4. Combine the results for the real and imaginary parts:

For the real part:
[tex]$$
64 + (-25) = 39.
$$[/tex]

The imaginary part remains:
[tex]$$
-80i.
$$[/tex]

Thus, the final result is:

[tex]$$
(8-5i)^2 = 39 - 80 i.
$$[/tex]