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Answer :
To solve the problem of multiplying and simplifying [tex]\((8-5i)^2\)[/tex], we can follow these steps:
1. Understand the Expression:
We need to square the complex number [tex]\((8 - 5i)\)[/tex].
2. Apply the Formula for Squaring a Binomial:
The formula for squaring a complex number [tex]\((a + bi)\)[/tex] is:
[tex]\[
(a + bi)^2 = a^2 + 2abi + (bi)^2
\][/tex]
Since in our case, [tex]\(b\)[/tex] is negative, we'll substitute [tex]\(-5i\)[/tex] for [tex]\(b\)[/tex].
3. Calculate Each Part:
- [tex]\(a^2\)[/tex]: Square the real part.
[tex]\[
a^2 = 8^2 = 64
\][/tex]
- [tex]\(2ab\)[/tex]: Calculate two times the product of the real and imaginary parts.
[tex]\[
2ab = 2 \times 8 \times (-5) = -80i
\][/tex]
- [tex]\((bi)^2\)[/tex]: Square the imaginary part, which includes squaring the imaginary unit [tex]\(i\)[/tex].
[tex]\[
(-5i)^2 = (-5)^2 \times (i)^2 = 25 \times (-1) = -25
\][/tex]
4. Combine Real and Imaginary Parts:
- Real Part: Sum of [tex]\(a^2\)[/tex] and the result from [tex]\((bi)^2\)[/tex].
[tex]\[
\text{Real Part} = 64 - 25 = 39
\][/tex]
- Imaginary Part: Directly from calculation of [tex]\(2ab\)[/tex].
[tex]\[
\text{Imaginary Part} = -80i
\][/tex]
5. Form the Final Answer:
Combine the real and imaginary results to express the final product:
[tex]\[
39 - 80i
\][/tex]
Thus, the correct product is [tex]\(\boxed{39 - 80i}\)[/tex].
1. Understand the Expression:
We need to square the complex number [tex]\((8 - 5i)\)[/tex].
2. Apply the Formula for Squaring a Binomial:
The formula for squaring a complex number [tex]\((a + bi)\)[/tex] is:
[tex]\[
(a + bi)^2 = a^2 + 2abi + (bi)^2
\][/tex]
Since in our case, [tex]\(b\)[/tex] is negative, we'll substitute [tex]\(-5i\)[/tex] for [tex]\(b\)[/tex].
3. Calculate Each Part:
- [tex]\(a^2\)[/tex]: Square the real part.
[tex]\[
a^2 = 8^2 = 64
\][/tex]
- [tex]\(2ab\)[/tex]: Calculate two times the product of the real and imaginary parts.
[tex]\[
2ab = 2 \times 8 \times (-5) = -80i
\][/tex]
- [tex]\((bi)^2\)[/tex]: Square the imaginary part, which includes squaring the imaginary unit [tex]\(i\)[/tex].
[tex]\[
(-5i)^2 = (-5)^2 \times (i)^2 = 25 \times (-1) = -25
\][/tex]
4. Combine Real and Imaginary Parts:
- Real Part: Sum of [tex]\(a^2\)[/tex] and the result from [tex]\((bi)^2\)[/tex].
[tex]\[
\text{Real Part} = 64 - 25 = 39
\][/tex]
- Imaginary Part: Directly from calculation of [tex]\(2ab\)[/tex].
[tex]\[
\text{Imaginary Part} = -80i
\][/tex]
5. Form the Final Answer:
Combine the real and imaginary results to express the final product:
[tex]\[
39 - 80i
\][/tex]
Thus, the correct product is [tex]\(\boxed{39 - 80i}\)[/tex].
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