Answer :
To solve [tex]\((8 - 5i)^2\)[/tex], we need to use the formula for squaring a binomial, [tex]\((a - bi)^2 = a^2 - 2abi + (bi)^2\)[/tex].
Here is a step-by-step process:
1. Identify the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- [tex]\(a = 8\)[/tex]
- [tex]\(b = 5\)[/tex]
2. Apply the formula to expand [tex]\((8 - 5i)^2\)[/tex]:
[tex]\[
(8 - 5i)^2 = a^2 - 2abi + (bi)^2
\][/tex]
3. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = 8^2 = 64
\][/tex]
4. Calculate [tex]\(-2abi\)[/tex]:
[tex]\[
-2abi = -2 \cdot 8 \cdot 5 \cdot i = -80i
\][/tex]
5. Calculate [tex]\((bi)^2\)[/tex]:
[tex]\[
(5i)^2 = 25i^2
\][/tex]
- Since [tex]\(i^2 = -1\)[/tex], we have:
[tex]\[
25i^2 = 25(-1) = -25
\][/tex]
6. Combine the results:
- Sum the real parts: [tex]\(64 - 25 = 39\)[/tex]
- The imaginary part is [tex]\(-80i\)[/tex]
Therefore, the simplified product is:
[tex]\[ 39 - 80i \][/tex]
So, the correct answer from the options given is [tex]\(\boxed{39 - 80i}\)[/tex].
Here is a step-by-step process:
1. Identify the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- [tex]\(a = 8\)[/tex]
- [tex]\(b = 5\)[/tex]
2. Apply the formula to expand [tex]\((8 - 5i)^2\)[/tex]:
[tex]\[
(8 - 5i)^2 = a^2 - 2abi + (bi)^2
\][/tex]
3. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = 8^2 = 64
\][/tex]
4. Calculate [tex]\(-2abi\)[/tex]:
[tex]\[
-2abi = -2 \cdot 8 \cdot 5 \cdot i = -80i
\][/tex]
5. Calculate [tex]\((bi)^2\)[/tex]:
[tex]\[
(5i)^2 = 25i^2
\][/tex]
- Since [tex]\(i^2 = -1\)[/tex], we have:
[tex]\[
25i^2 = 25(-1) = -25
\][/tex]
6. Combine the results:
- Sum the real parts: [tex]\(64 - 25 = 39\)[/tex]
- The imaginary part is [tex]\(-80i\)[/tex]
Therefore, the simplified product is:
[tex]\[ 39 - 80i \][/tex]
So, the correct answer from the options given is [tex]\(\boxed{39 - 80i}\)[/tex].
