Answer :
To solve the problem of multiplying and simplifying [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]
In this expression, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex]. Let's go through the steps:
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
8^2 = 64.
\][/tex]
2. Calculate [tex]\(-2ab\)[/tex]:
[tex]\[
-2 \cdot 8 \cdot 5i = -80i.
\][/tex]
3. Calculate [tex]\(b^2\)[/tex] (considering [tex]\(i^2 = -1\)[/tex]):
[tex]\[
(5i)^2 = 25i^2.
\][/tex]
Since [tex]\(i^2 = -1\)[/tex], we have:
[tex]\[
25 \cdot (-1) = -25.
\][/tex]
4. Combine the terms:
- The real part is [tex]\(64 - 25 = 39\)[/tex].
- The imaginary part is [tex]\(-80i\)[/tex].
Therefore, the simplified product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex].
The correct option that matches our solution is [tex]\(89 - 80i\)[/tex]. However, please note that there might have been an error in matching options, so ensure to select the answer [tex]\(39 - 80i\)[/tex] based on the calculations. If the correct match isn't present, double-check your provided options or the context in which they are being presented.
[tex]\[(a - b)^2 = a^2 - 2ab + b^2.\][/tex]
In this expression, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex]. Let's go through the steps:
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
8^2 = 64.
\][/tex]
2. Calculate [tex]\(-2ab\)[/tex]:
[tex]\[
-2 \cdot 8 \cdot 5i = -80i.
\][/tex]
3. Calculate [tex]\(b^2\)[/tex] (considering [tex]\(i^2 = -1\)[/tex]):
[tex]\[
(5i)^2 = 25i^2.
\][/tex]
Since [tex]\(i^2 = -1\)[/tex], we have:
[tex]\[
25 \cdot (-1) = -25.
\][/tex]
4. Combine the terms:
- The real part is [tex]\(64 - 25 = 39\)[/tex].
- The imaginary part is [tex]\(-80i\)[/tex].
Therefore, the simplified product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex].
The correct option that matches our solution is [tex]\(89 - 80i\)[/tex]. However, please note that there might have been an error in matching options, so ensure to select the answer [tex]\(39 - 80i\)[/tex] based on the calculations. If the correct match isn't present, double-check your provided options or the context in which they are being presented.
