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Answer :
To multiply and simplify the product of [tex]\((8 - 5i)^2\)[/tex], we can follow these steps:
1. Understand the Expression: We want to square the complex number [tex]\(8 - 5i\)[/tex].
2. Expand the Square: Use the formula for the square of a binomial, [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].
- Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
3. Calculate Each Part:
- [tex]\(a^2 = 8^2 = 64\)[/tex]
- [tex]\(b^2 = (5i)^2 = 25i^2 = 25(-1) = -25\)[/tex] (since [tex]\(i^2 = -1\)[/tex])
- [tex]\(-2ab = -2 \times 8 \times 5i = -80i\)[/tex]
4. Combine the Results:
- Add the results together: [tex]\(64 - 80i - 25\)[/tex].
5. Simplify:
- Combine the real parts: [tex]\(64 - 25 = 39\)[/tex].
- The imaginary part remains [tex]\(-80i\)[/tex].
6. Final Answer: The simplified form of [tex]\((8-5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex].
So, the correct choice is [tex]\(39 - 80i\)[/tex].
1. Understand the Expression: We want to square the complex number [tex]\(8 - 5i\)[/tex].
2. Expand the Square: Use the formula for the square of a binomial, [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].
- Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
3. Calculate Each Part:
- [tex]\(a^2 = 8^2 = 64\)[/tex]
- [tex]\(b^2 = (5i)^2 = 25i^2 = 25(-1) = -25\)[/tex] (since [tex]\(i^2 = -1\)[/tex])
- [tex]\(-2ab = -2 \times 8 \times 5i = -80i\)[/tex]
4. Combine the Results:
- Add the results together: [tex]\(64 - 80i - 25\)[/tex].
5. Simplify:
- Combine the real parts: [tex]\(64 - 25 = 39\)[/tex].
- The imaginary part remains [tex]\(-80i\)[/tex].
6. Final Answer: The simplified form of [tex]\((8-5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex].
So, the correct choice is [tex]\(39 - 80i\)[/tex].
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