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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 product, we first write the expression as the square of the complex number:

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

Now, we use the distributive property (also known as the FOIL method) to expand the product:

1. Multiply the first terms:
[tex]$$
8 \times 8 = 64.
$$[/tex]

2. Multiply the outer terms:
[tex]$$
8 \times (-5i) = -40i.
$$[/tex]

3. Multiply the inner terms:
[tex]$$
(-5i) \times 8 = -40i.
$$[/tex]

4. Multiply the last terms:
[tex]$$
(-5i) \times (-5i) = 25i^2.
$$[/tex]

Since [tex]$i^2 = -1$[/tex], we have:

[tex]$$
25i^2 = 25(-1) = -25.
$$[/tex]

Next, combine all the parts:

[tex]$$
64 - 40i - 40i - 25.
$$[/tex]

Combine the like terms:

- The real parts: [tex]$64 - 25 = 39$[/tex].
- The imaginary parts: [tex]$-40i - 40i = -80i$[/tex].

Thus, the simplified product is:

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

So, the final answer is [tex]$\boxed{39-80i}$[/tex].

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