Answer :
To multiply and simplify the expression [tex]\((8 - 5i)^2\)[/tex], we need to apply the formula for the square of a binomial:
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]
For the expression [tex]\((8 - 5i)^2\)[/tex], set [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex]. Plug these values into the formula:
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
8^2 = 64
\][/tex]
2. Calculate [tex]\(-2ab\)[/tex]:
[tex]\[
-2 \times 8 \times 5i = -80i
\][/tex]
3. Calculate [tex]\(b^2\)[/tex]:
[tex]\[
(5i)^2 = 25i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex], this becomes:
[tex]\[
25 \times -1 = -25
\][/tex]
Now, combine all these results together:
[tex]\[
(8 - 5i)^2 = 64 - 80i - 25
\][/tex]
Simplify by combining the real terms:
[tex]\[
64 - 25 = 39
\][/tex]
Thus, the simplified expression is:
[tex]\[
39 - 80i
\][/tex]
So, the correct answer is [tex]\(39 - 80i\)[/tex].
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]
For the expression [tex]\((8 - 5i)^2\)[/tex], set [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex]. Plug these values into the formula:
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
8^2 = 64
\][/tex]
2. Calculate [tex]\(-2ab\)[/tex]:
[tex]\[
-2 \times 8 \times 5i = -80i
\][/tex]
3. Calculate [tex]\(b^2\)[/tex]:
[tex]\[
(5i)^2 = 25i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex], this becomes:
[tex]\[
25 \times -1 = -25
\][/tex]
Now, combine all these results together:
[tex]\[
(8 - 5i)^2 = 64 - 80i - 25
\][/tex]
Simplify by combining the real terms:
[tex]\[
64 - 25 = 39
\][/tex]
Thus, the simplified expression is:
[tex]\[
39 - 80i
\][/tex]
So, the correct answer is [tex]\(39 - 80i\)[/tex].
