Answer :
To multiply and simplify the expression [tex]\((8 - 5i)^2\)[/tex], follow these steps:
1. Express the expression:
[tex]\((8 - 5i)^2\)[/tex] means we're multiplying [tex]\((8 - 5i)\)[/tex] by itself.
2. Use the formula for the square of a binomial:
[tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].
In this case, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
3. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = 8^2 = 64
\][/tex]
4. Calculate [tex]\(-2ab\)[/tex]:
[tex]\[
-2ab = -2 \times 8 \times 5i = -80i
\][/tex]
5. Calculate [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = (5i)^2 = 25i^2
\][/tex]
Remember, [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
b^2 = 25(-1) = -25
\][/tex]
6. Combine the results:
Combine the results from steps 3, 4, and 5:
[tex]\[
(8 - 5i)^2 = 64 - 80i - 25
\][/tex]
7. Simplify the expression:
Combine the real parts:
[tex]\[
64 - 25 = 39
\][/tex]
So the expression becomes:
[tex]\[
39 - 80i
\][/tex]
Thus, the simplified product of [tex]\((8-5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex].
From the given options, the correct choice is [tex]\(39 - 80i\)[/tex].
1. Express the expression:
[tex]\((8 - 5i)^2\)[/tex] means we're multiplying [tex]\((8 - 5i)\)[/tex] by itself.
2. Use the formula for the square of a binomial:
[tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].
In this case, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
3. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = 8^2 = 64
\][/tex]
4. Calculate [tex]\(-2ab\)[/tex]:
[tex]\[
-2ab = -2 \times 8 \times 5i = -80i
\][/tex]
5. Calculate [tex]\(b^2\)[/tex]:
[tex]\[
b^2 = (5i)^2 = 25i^2
\][/tex]
Remember, [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
b^2 = 25(-1) = -25
\][/tex]
6. Combine the results:
Combine the results from steps 3, 4, and 5:
[tex]\[
(8 - 5i)^2 = 64 - 80i - 25
\][/tex]
7. Simplify the expression:
Combine the real parts:
[tex]\[
64 - 25 = 39
\][/tex]
So the expression becomes:
[tex]\[
39 - 80i
\][/tex]
Thus, the simplified product of [tex]\((8-5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex].
From the given options, the correct choice is [tex]\(39 - 80i\)[/tex].
