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Answer :
To multiply and simplify the product of [tex]\((8 - 5i)^2\)[/tex], follow these steps:
1. Set up the expression:
We need to square the complex number: [tex]\((8 - 5i) \times (8 - 5i)\)[/tex].
2. Apply the distributive property (FOIL method):
Expand the expression:
[tex]\[
(8 - 5i)(8 - 5i) = 8 \times 8 + 8 \times (-5i) + (-5i) \times 8 + (-5i) \times (-5i)
\][/tex]
3. Calculate each part:
- [tex]\(8 \times 8 = 64\)[/tex]
- [tex]\(8 \times (-5i) = -40i\)[/tex]
- [tex]\((-5i) \times 8 = -40i\)[/tex]
- [tex]\((-5i) \times (-5i) = 25i^2\)[/tex]
4. Simplify the imaginary part:
Recall that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
25i^2 = 25(-1) = -25
\][/tex]
5. Combine all parts:
Add the real and imaginary components:
[tex]\[
64 - 25 + (-40i) + (-40i) = 39 - 80i
\][/tex]
Thus, when you multiply and simplify [tex]\((8 - 5i)^2\)[/tex], the product is [tex]\(39 - 80i\)[/tex].
The correct answer from the given options is [tex]\(39 - 80i\)[/tex].
1. Set up the expression:
We need to square the complex number: [tex]\((8 - 5i) \times (8 - 5i)\)[/tex].
2. Apply the distributive property (FOIL method):
Expand the expression:
[tex]\[
(8 - 5i)(8 - 5i) = 8 \times 8 + 8 \times (-5i) + (-5i) \times 8 + (-5i) \times (-5i)
\][/tex]
3. Calculate each part:
- [tex]\(8 \times 8 = 64\)[/tex]
- [tex]\(8 \times (-5i) = -40i\)[/tex]
- [tex]\((-5i) \times 8 = -40i\)[/tex]
- [tex]\((-5i) \times (-5i) = 25i^2\)[/tex]
4. Simplify the imaginary part:
Recall that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
25i^2 = 25(-1) = -25
\][/tex]
5. Combine all parts:
Add the real and imaginary components:
[tex]\[
64 - 25 + (-40i) + (-40i) = 39 - 80i
\][/tex]
Thus, when you multiply and simplify [tex]\((8 - 5i)^2\)[/tex], the product is [tex]\(39 - 80i\)[/tex].
The correct answer from the given options is [tex]\(39 - 80i\)[/tex].
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