Answer :
To multiply and simplify the expression [tex]\((8 - 5i)^2\)[/tex], let's begin by using the formula for squaring a binomial: [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].
For the expression [tex]\((8 - 5i)^2\)[/tex]:
1. Identify the terms:
- Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
2. Apply the formula:
[tex]\[
(8 - 5i)^2 = 8^2 - 2 \times 8 \times 5i + (5i)^2
\][/tex]
3. Calculate each part:
- [tex]\(8^2 = 64\)[/tex]
- [tex]\(-2 \times 8 \times 5i = -80i\)[/tex]
- [tex]\((5i)^2 = 25i^2\)[/tex]. Since [tex]\(i^2 = -1\)[/tex], this term becomes [tex]\(25 \times -1 = -25\)[/tex].
4. Combine the results:
- The real part: [tex]\(64 - 25 = 39\)[/tex]
- The imaginary part: [tex]\(-80i\)[/tex]
Combine these to get the final simplified form of the expression:
[tex]\[
39 - 80i
\][/tex]
Thus, the product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex]. Among the given choices, the correct answer is [tex]\(\boxed{39 - 80i}\)[/tex].
For the expression [tex]\((8 - 5i)^2\)[/tex]:
1. Identify the terms:
- Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
2. Apply the formula:
[tex]\[
(8 - 5i)^2 = 8^2 - 2 \times 8 \times 5i + (5i)^2
\][/tex]
3. Calculate each part:
- [tex]\(8^2 = 64\)[/tex]
- [tex]\(-2 \times 8 \times 5i = -80i\)[/tex]
- [tex]\((5i)^2 = 25i^2\)[/tex]. Since [tex]\(i^2 = -1\)[/tex], this term becomes [tex]\(25 \times -1 = -25\)[/tex].
4. Combine the results:
- The real part: [tex]\(64 - 25 = 39\)[/tex]
- The imaginary part: [tex]\(-80i\)[/tex]
Combine these to get the final simplified form of the expression:
[tex]\[
39 - 80i
\][/tex]
Thus, the product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex]. Among the given choices, the correct answer is [tex]\(\boxed{39 - 80i}\)[/tex].
