College

Multiply and simplify the product:

[tex](8-5i)^2[/tex]

Select the product:

A. 39
B. 89
C. [tex]39 - 80i[/tex]
D. [tex]89 - 80i[/tex]

Answer :

To solve the problem of multiplying and simplifying the product [tex]\((8 - 5i)^2\)[/tex], follow these steps:

1. Expand the expression [tex]\((8 - 5i)^2\)[/tex] using the formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]:
[tex]\[
(8 - 5i)^2 = 8^2 - 2 \cdot 8 \cdot 5i + (5i)^2
\][/tex]

2. Calculate each part separately:
- [tex]\(8^2 = 64\)[/tex]
- [tex]\(-2 \cdot 8 \cdot 5i = -80i\)[/tex]
- [tex]\((5i)^2 = 25i^2\)[/tex]

3. Remember that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
(5i)^2 = 25i^2 = 25(-1) = -25
\][/tex]

4. Combine all the parts together:
[tex]\[
64 - 80i - 25
\][/tex]

5. Simplify by combining the real parts (64 and -25):
[tex]\[
64 - 25 = 39
\][/tex]

6. Write the simplified expression:
[tex]\[
39 - 80i
\][/tex]

So, the product is [tex]\(\boxed{39 - 80i}\)[/tex].