Answer :
To multiply and simplify the expression [tex]\((8 - 5i)^2\)[/tex], we can use the formula for squaring a binomial: [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].
Here are the steps:
1. Identify the terms:
- The first term [tex]\(a\)[/tex] is 8.
- The second term [tex]\(b\)[/tex] is [tex]\(5i\)[/tex].
2. Square the first term ([tex]\(a^2\)[/tex]):
[tex]\[
8^2 = 64
\][/tex]
3. Multiply the terms and double it ([tex]\(-2ab\)[/tex]):
[tex]\[
-2 \times 8 \times 5i = -80i
\][/tex]
4. Square the second term ([tex]\(b^2\)[/tex]):
- Since [tex]\(i^2 = -1\)[/tex], we calculate [tex]\((5i)^2\)[/tex] as follows:
[tex]\[
(5i)^2 = 25i^2 = 25 \times (-1) = -25
\][/tex]
5. Combine the results:
- Add the results from steps 2, 3, and 4:
[tex]\[
64 - 80i - 25
\][/tex]
6. Simplify the expression:
- Combine the real parts: [tex]\(64 - 25 = 39\)[/tex].
- Combine the imaginary part: [tex]\(-80i\)[/tex].
So, the simplified form of the expression is:
[tex]\[
39 - 80i
\][/tex]
Therefore, the correct product is [tex]\(\boxed{39 - 80i}\)[/tex].
Here are the steps:
1. Identify the terms:
- The first term [tex]\(a\)[/tex] is 8.
- The second term [tex]\(b\)[/tex] is [tex]\(5i\)[/tex].
2. Square the first term ([tex]\(a^2\)[/tex]):
[tex]\[
8^2 = 64
\][/tex]
3. Multiply the terms and double it ([tex]\(-2ab\)[/tex]):
[tex]\[
-2 \times 8 \times 5i = -80i
\][/tex]
4. Square the second term ([tex]\(b^2\)[/tex]):
- Since [tex]\(i^2 = -1\)[/tex], we calculate [tex]\((5i)^2\)[/tex] as follows:
[tex]\[
(5i)^2 = 25i^2 = 25 \times (-1) = -25
\][/tex]
5. Combine the results:
- Add the results from steps 2, 3, and 4:
[tex]\[
64 - 80i - 25
\][/tex]
6. Simplify the expression:
- Combine the real parts: [tex]\(64 - 25 = 39\)[/tex].
- Combine the imaginary part: [tex]\(-80i\)[/tex].
So, the simplified form of the expression is:
[tex]\[
39 - 80i
\][/tex]
Therefore, the correct product is [tex]\(\boxed{39 - 80i}\)[/tex].
