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].
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].
