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Answer :
To multiply and simplify the expression [tex]\((8 - 5i)^2\)[/tex], you can follow these steps:
1. Express the Problem: We're asked to find the square of the complex number [tex]\(8 - 5i\)[/tex].
2. Apply the Formula: Use the formula for squaring 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]\(-2ab = -2 \times 8 \times 5i = -80i\)[/tex]
- [tex]\(b^2 = (5i)^2 = 25i^2\)[/tex]. Since [tex]\(i^2 = -1\)[/tex], [tex]\(25i^2 = 25 \times (-1) = -25\)[/tex].
4. Combine the Results:
- Now, substitute these values back into the formula:
[tex]\[
(8 - 5i)^2 = 64 - 80i - 25
\][/tex]
5. Combine Like Terms:
- Combine the real parts: [tex]\(64 - 25 = 39\)[/tex]
- The imaginary part remains [tex]\(-80i\)[/tex].
6. Write the Final Result:
- Therefore, the simplified product is [tex]\(39 - 80i\)[/tex].
Hence, the correct product is [tex]\(39 - 80i\)[/tex].
1. Express the Problem: We're asked to find the square of the complex number [tex]\(8 - 5i\)[/tex].
2. Apply the Formula: Use the formula for squaring 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]\(-2ab = -2 \times 8 \times 5i = -80i\)[/tex]
- [tex]\(b^2 = (5i)^2 = 25i^2\)[/tex]. Since [tex]\(i^2 = -1\)[/tex], [tex]\(25i^2 = 25 \times (-1) = -25\)[/tex].
4. Combine the Results:
- Now, substitute these values back into the formula:
[tex]\[
(8 - 5i)^2 = 64 - 80i - 25
\][/tex]
5. Combine Like Terms:
- Combine the real parts: [tex]\(64 - 25 = 39\)[/tex]
- The imaginary part remains [tex]\(-80i\)[/tex].
6. Write the Final Result:
- Therefore, the simplified product is [tex]\(39 - 80i\)[/tex].
Hence, the correct product is [tex]\(39 - 80i\)[/tex].
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