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Answer :
To solve the problem of multiplying and simplifying the product of [tex]\((8 - 5i)^2\)[/tex], we need to follow these steps:
1. Identify the expression: We have the complex number [tex]\((8 - 5i)\)[/tex].
2. Apply the formula for squaring a binomial: We will use the formula [tex]\((a - bi)^2 = a^2 - 2abi + (bi)^2\)[/tex].
3. Plug in the values:
- Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5\)[/tex] (since it's [tex]\(-5i\)[/tex], we take [tex]\(b = 5\)[/tex] for calculation).
- The expression becomes:
[tex]\((8)^2 - 2 \times 8 \times 5i + (5i)^2\)[/tex].
4. Calculate each part:
- Calculate [tex]\(a^2 = 8^2 = 64\)[/tex].
- Calculate [tex]\(-2ab \times i = -2 \times 8 \times 5 \times i = -80i\)[/tex].
- Calculate [tex]\((bi)^2 = (5i)^2 = 25i^2\)[/tex].
5. Substitute [tex]\(i^2 = -1\)[/tex]:
- The term [tex]\(25i^2\)[/tex] becomes [tex]\(25 \times (-1) = -25\)[/tex].
6. Combine the real and imaginary parts:
- Real part: [tex]\(64 + (-25) = 39\)[/tex].
- Imaginary part: [tex]\(-80i\)[/tex].
7. Write the final result:
- The simplified product is [tex]\(39 - 80i\)[/tex].
Therefore, the product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex]. The correct selection from the given options is 39 - 80i.
1. Identify the expression: We have the complex number [tex]\((8 - 5i)\)[/tex].
2. Apply the formula for squaring a binomial: We will use the formula [tex]\((a - bi)^2 = a^2 - 2abi + (bi)^2\)[/tex].
3. Plug in the values:
- Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5\)[/tex] (since it's [tex]\(-5i\)[/tex], we take [tex]\(b = 5\)[/tex] for calculation).
- The expression becomes:
[tex]\((8)^2 - 2 \times 8 \times 5i + (5i)^2\)[/tex].
4. Calculate each part:
- Calculate [tex]\(a^2 = 8^2 = 64\)[/tex].
- Calculate [tex]\(-2ab \times i = -2 \times 8 \times 5 \times i = -80i\)[/tex].
- Calculate [tex]\((bi)^2 = (5i)^2 = 25i^2\)[/tex].
5. Substitute [tex]\(i^2 = -1\)[/tex]:
- The term [tex]\(25i^2\)[/tex] becomes [tex]\(25 \times (-1) = -25\)[/tex].
6. Combine the real and imaginary parts:
- Real part: [tex]\(64 + (-25) = 39\)[/tex].
- Imaginary part: [tex]\(-80i\)[/tex].
7. Write the final result:
- The simplified product is [tex]\(39 - 80i\)[/tex].
Therefore, the product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex]. The correct selection from the given options is 39 - 80i.
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