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Answer :
To find the product and simplify [tex]\((8 - 5i)^2\)[/tex], we will use the formula for squaring a binomial:
[tex]\[
(a - bi)^2 = a^2 - 2abi + (bi)^2
\][/tex]
In this case, [tex]\(a = 8\)[/tex] and [tex]\(b = 5\)[/tex]. Let's go step-by-step:
1. Square the real part ([tex]\(a\)[/tex]):
[tex]\[
a^2 = 8^2 = 64
\][/tex]
2. Multiply the real part and the imaginary unit, and then double it ([tex]\(-2ab\)[/tex]):
[tex]\[
2ab = 2 \times 8 \times 5 = 80
\][/tex]
Since there is a negative in front of [tex]\(b\)[/tex], this makes it [tex]\(-80i\)[/tex].
3. Square the imaginary part ([tex]\((bi)^2\)[/tex]):
[tex]\[
(5i)^2 = (-5)^2 \times i^2 = 25 \times (-1) = -25
\][/tex]
Here, [tex]\(i^2 = -1\)[/tex].
4. Combine all parts to form the simplified expression:
[tex]\[
(8 - 5i)^2 = 64 - 80i - 25
\][/tex]
[tex]\[
= (64 - 25) - 80i
\][/tex]
[tex]\[
= 39 - 80i
\][/tex]
Therefore, the product [tex]\((8-5i)^2\)[/tex] simplifies to [tex]\(39 - 80i\)[/tex].
The correct selection from the given options is [tex]\(\boxed{39 - 80i}\)[/tex].
[tex]\[
(a - bi)^2 = a^2 - 2abi + (bi)^2
\][/tex]
In this case, [tex]\(a = 8\)[/tex] and [tex]\(b = 5\)[/tex]. Let's go step-by-step:
1. Square the real part ([tex]\(a\)[/tex]):
[tex]\[
a^2 = 8^2 = 64
\][/tex]
2. Multiply the real part and the imaginary unit, and then double it ([tex]\(-2ab\)[/tex]):
[tex]\[
2ab = 2 \times 8 \times 5 = 80
\][/tex]
Since there is a negative in front of [tex]\(b\)[/tex], this makes it [tex]\(-80i\)[/tex].
3. Square the imaginary part ([tex]\((bi)^2\)[/tex]):
[tex]\[
(5i)^2 = (-5)^2 \times i^2 = 25 \times (-1) = -25
\][/tex]
Here, [tex]\(i^2 = -1\)[/tex].
4. Combine all parts to form the simplified expression:
[tex]\[
(8 - 5i)^2 = 64 - 80i - 25
\][/tex]
[tex]\[
= (64 - 25) - 80i
\][/tex]
[tex]\[
= 39 - 80i
\][/tex]
Therefore, the product [tex]\((8-5i)^2\)[/tex] simplifies to [tex]\(39 - 80i\)[/tex].
The correct selection from the given options is [tex]\(\boxed{39 - 80i}\)[/tex].
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