Answer :
To solve the expression [tex]\((8 - 5i)^2\)[/tex], we can use the formula for the square of a binomial:
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]
Let's identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] from the expression:
- [tex]\(a = 8\)[/tex]
- [tex]\(b = 5i\)[/tex]
Now, let's apply the formula step-by-step:
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = 8^2 = 64
\][/tex]
2. Calculate [tex]\(-2ab\)[/tex]:
[tex]\[
-2ab = -2 \times 8 \times (5i) = -80i
\][/tex]
3. Calculate [tex]\(b^2\)[/tex]:
Since [tex]\(b = 5i\)[/tex], then:
[tex]\[
b^2 = (5i)^2 = 25i^2
\][/tex]
We know that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
25i^2 = 25 \times -1 = -25
\][/tex]
4. Combine the results:
Now, summing up all the parts from the formula:
[tex]\[
(8 - 5i)^2 = 64 + (-80i) + (-25)
\][/tex]
Simplify it further:
[tex]\[
= 64 - 25 - 80i
\][/tex]
[tex]\[
= 39 - 80i
\][/tex]
Thus, the simplified product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex]. Therefore, the answer is [tex]\(\boxed{39 - 80i}\)[/tex].
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]
Let's identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] from the expression:
- [tex]\(a = 8\)[/tex]
- [tex]\(b = 5i\)[/tex]
Now, let's apply the formula step-by-step:
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = 8^2 = 64
\][/tex]
2. Calculate [tex]\(-2ab\)[/tex]:
[tex]\[
-2ab = -2 \times 8 \times (5i) = -80i
\][/tex]
3. Calculate [tex]\(b^2\)[/tex]:
Since [tex]\(b = 5i\)[/tex], then:
[tex]\[
b^2 = (5i)^2 = 25i^2
\][/tex]
We know that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
25i^2 = 25 \times -1 = -25
\][/tex]
4. Combine the results:
Now, summing up all the parts from the formula:
[tex]\[
(8 - 5i)^2 = 64 + (-80i) + (-25)
\][/tex]
Simplify it further:
[tex]\[
= 64 - 25 - 80i
\][/tex]
[tex]\[
= 39 - 80i
\][/tex]
Thus, the simplified product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex]. Therefore, the answer is [tex]\(\boxed{39 - 80i}\)[/tex].
