Answer :
To solve the problem of multiplying and simplifying the product [tex]\((8 - 5i)^2\)[/tex], we can follow these steps:
1. Express the problem: Start with [tex]\((8 - 5i)^2\)[/tex].
2. Use the formula for squaring a binomial:
[tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].
3. Identify the values: Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
4. Calculate each part:
- Calculate [tex]\(a^2\)[/tex]:
[tex]\[8^2 = 64\][/tex]
- Calculate [tex]\(-2ab\)[/tex]:
[tex]\[-2 \times 8 \times 5i = -80i\][/tex]
- Calculate [tex]\(b^2\)[/tex]:
[tex]\((5i)^2 = 25i^2 = 25 \times (-1) = -25\]
(since \(i^2 = -1\)[/tex])
5. Combine the results:
[tex]\[
(8 - 5i)^2 = a^2 - 2ab + b^2 = 64 - 80i - 25
\][/tex]
6. Simplify the expression:
[tex]\[
64 - 25 - 80i = 39 - 80i
\][/tex]
Therefore, the simplified product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex].
Thus, the correct answer is [tex]\(39 - 80i\)[/tex].
1. Express the problem: Start with [tex]\((8 - 5i)^2\)[/tex].
2. Use the formula for squaring a binomial:
[tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].
3. Identify the values: Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
4. Calculate each part:
- Calculate [tex]\(a^2\)[/tex]:
[tex]\[8^2 = 64\][/tex]
- Calculate [tex]\(-2ab\)[/tex]:
[tex]\[-2 \times 8 \times 5i = -80i\][/tex]
- Calculate [tex]\(b^2\)[/tex]:
[tex]\((5i)^2 = 25i^2 = 25 \times (-1) = -25\]
(since \(i^2 = -1\)[/tex])
5. Combine the results:
[tex]\[
(8 - 5i)^2 = a^2 - 2ab + b^2 = 64 - 80i - 25
\][/tex]
6. Simplify the expression:
[tex]\[
64 - 25 - 80i = 39 - 80i
\][/tex]
Therefore, the simplified product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex].
Thus, the correct answer is [tex]\(39 - 80i\)[/tex].
