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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. [tex]$89 - 80i$[/tex]

Answer :

To find the product of [tex]\((8 - 5i)^2\)[/tex] and simplify it, we can use the formula for the square of a binomial:

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

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

Step 1: Calculate [tex]\(a^2\)[/tex].

[tex]\[
a^2 = 8^2 = 64
\][/tex]

Step 2: Calculate [tex]\(2ab\)[/tex].

[tex]\[
2ab = 2 \times 8 \times 5i = 80i
\][/tex]

Step 3: Calculate [tex]\(b^2\)[/tex]. Remember that [tex]\(b = 5i\)[/tex], so:

[tex]\[
b^2 = (5i)^2 = 25i^2
\][/tex]

Since [tex]\(i^2 = -1\)[/tex], we have:

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

Step 4: Substitute these values into the binomial formula:

[tex]\[
(8 - 5i)^2 = 64 - 80i - 25
\][/tex]

Step 5: Combine like terms to simplify:

Real parts: [tex]\(64 - 25 = 39\)[/tex]

Imaginary part: [tex]\(-80i\)[/tex]

So, the simplified product is:

[tex]\[
39 - 80i
\][/tex]

The correct answer is [tex]\(39 - 80i\)[/tex].

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