Answer :
Sure! Let's go through the process of multiplying and simplifying [tex]\((8 - 5i)^2\)[/tex] step-by-step.
1. Understand the Expression:
You need to square the complex number [tex]\(8 - 5i\)[/tex]. This means you are multiplying the expression by itself: [tex]\((8 - 5i)(8 - 5i)\)[/tex].
2. Use the Formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]:
Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
3. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
(8)^2 = 64
\][/tex]
4. Calculate [tex]\(-2ab\)[/tex]:
[tex]\[
-2 \times 8 \times (-5i) = 80i
\][/tex]
Note the negative sign in front of [tex]\(b\)[/tex] and the imaginary unit [tex]\(i\)[/tex].
5. Calculate [tex]\(b^2\)[/tex]:
[tex]\[
(-5i)^2 = 25i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex], it becomes:
[tex]\[
25(-1) = -25
\][/tex]
6. Combine the Results:
[tex]\[
a^2 - 2ab + b^2 = 64 + 80i - 25
\][/tex]
7. Simplify the Result:
Combine the real parts and the imaginary part:
[tex]\[
64 - 25 = 39
\][/tex]
Thus, the expression becomes:
[tex]\[
39 + 80i
\][/tex]
The simplified product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(89 - 80i\)[/tex].
So, the correct option is 89-80i.
1. Understand the Expression:
You need to square the complex number [tex]\(8 - 5i\)[/tex]. This means you are multiplying the expression by itself: [tex]\((8 - 5i)(8 - 5i)\)[/tex].
2. Use the Formula [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]:
Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
3. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
(8)^2 = 64
\][/tex]
4. Calculate [tex]\(-2ab\)[/tex]:
[tex]\[
-2 \times 8 \times (-5i) = 80i
\][/tex]
Note the negative sign in front of [tex]\(b\)[/tex] and the imaginary unit [tex]\(i\)[/tex].
5. Calculate [tex]\(b^2\)[/tex]:
[tex]\[
(-5i)^2 = 25i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex], it becomes:
[tex]\[
25(-1) = -25
\][/tex]
6. Combine the Results:
[tex]\[
a^2 - 2ab + b^2 = 64 + 80i - 25
\][/tex]
7. Simplify the Result:
Combine the real parts and the imaginary part:
[tex]\[
64 - 25 = 39
\][/tex]
Thus, the expression becomes:
[tex]\[
39 + 80i
\][/tex]
The simplified product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(89 - 80i\)[/tex].
So, the correct option is 89-80i.
