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Answer :
To solve [tex]\((8 - 5i)^2\)[/tex], we want to expand and simplify this expression. Here's a step-by-step guide:
1. Recognize the expression: We have [tex]\((8 - 5i)^2\)[/tex]. This is a complex number, squared.
2. Use the formula for squaring a binomial:
[tex]\[
(a - bi)^2 = a^2 - 2abi + (bi)^2
\][/tex]
where [tex]\(a = 8\)[/tex] and [tex]\(b = 5\)[/tex].
3. Calculate each part:
- [tex]\(a^2 = 8^2 = 64\)[/tex]
- [tex]\(2ab = 2 \times 8 \times 5 = 80\)[/tex]. Since it's [tex]\(8 - 5i\)[/tex], this term will be negative: [tex]\(-80i\)[/tex].
- [tex]\((bi)^2 = (5i)^2 = 25i^2\)[/tex]. Remember, [tex]\(i^2 = -1\)[/tex], so [tex]\(25i^2 = 25 \times -1 = -25\)[/tex].
4. Combine the terms:
- The real part: [tex]\(a^2 + (bi)^2 = 64 - 25 = 39\)[/tex]
- The imaginary part: [tex]\(-80i\)[/tex]
Therefore, the simplified product is [tex]\(39 - 80i\)[/tex].
So, the correct option from the list is [tex]\(\boxed{39 - 80i}\)[/tex].
1. Recognize the expression: We have [tex]\((8 - 5i)^2\)[/tex]. This is a complex number, squared.
2. Use the formula for squaring a binomial:
[tex]\[
(a - bi)^2 = a^2 - 2abi + (bi)^2
\][/tex]
where [tex]\(a = 8\)[/tex] and [tex]\(b = 5\)[/tex].
3. Calculate each part:
- [tex]\(a^2 = 8^2 = 64\)[/tex]
- [tex]\(2ab = 2 \times 8 \times 5 = 80\)[/tex]. Since it's [tex]\(8 - 5i\)[/tex], this term will be negative: [tex]\(-80i\)[/tex].
- [tex]\((bi)^2 = (5i)^2 = 25i^2\)[/tex]. Remember, [tex]\(i^2 = -1\)[/tex], so [tex]\(25i^2 = 25 \times -1 = -25\)[/tex].
4. Combine the terms:
- The real part: [tex]\(a^2 + (bi)^2 = 64 - 25 = 39\)[/tex]
- The imaginary part: [tex]\(-80i\)[/tex]
Therefore, the simplified product is [tex]\(39 - 80i\)[/tex].
So, the correct option from the list is [tex]\(\boxed{39 - 80i}\)[/tex].
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