Answer :
To multiply and simplify [tex]\((8 - 5i)^2\)[/tex], we can follow these steps:
1. Understand the expression: We need to find the square of the complex number [tex]\(8 - 5i\)[/tex].
2. Use the formula for squaring a binomial:
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]
Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
3. Apply the formula:
[tex]\[
(8 - 5i)^2 = 8^2 - 2 \cdot 8 \cdot 5i + (5i)^2
\][/tex]
4. Calculate each part:
- [tex]\(8^2 = 64\)[/tex]
- [tex]\(2 \cdot 8 \cdot 5i = 80i\)[/tex]
- [tex]\((5i)^2 = 25i^2\)[/tex]
5. Simplify using [tex]\(i^2 = -1\)[/tex]:
- [tex]\(25i^2 = 25 \times -1 = -25\)[/tex]
6. Combine all the parts:
[tex]\[
64 - 25 - 80i = 39 - 80i
\][/tex]
So, the product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex].
Therefore, the selected product is [tex]\(\boxed{39 - 80i}\)[/tex].
1. Understand the expression: We need to find the square of the complex number [tex]\(8 - 5i\)[/tex].
2. Use the formula for squaring a binomial:
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]
Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
3. Apply the formula:
[tex]\[
(8 - 5i)^2 = 8^2 - 2 \cdot 8 \cdot 5i + (5i)^2
\][/tex]
4. Calculate each part:
- [tex]\(8^2 = 64\)[/tex]
- [tex]\(2 \cdot 8 \cdot 5i = 80i\)[/tex]
- [tex]\((5i)^2 = 25i^2\)[/tex]
5. Simplify using [tex]\(i^2 = -1\)[/tex]:
- [tex]\(25i^2 = 25 \times -1 = -25\)[/tex]
6. Combine all the parts:
[tex]\[
64 - 25 - 80i = 39 - 80i
\][/tex]
So, the product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex].
Therefore, the selected product is [tex]\(\boxed{39 - 80i}\)[/tex].
