Answer :
To multiply and simplify the expression [tex]\((8 - 5i)^2\)[/tex], we can follow these steps:
1. Write the expression being multiplied:
[tex]\[
(8 - 5i) \times (8 - 5i)
\][/tex]
2. 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 term individually:
- [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], it becomes [tex]\(25 \times -1 = -25\)[/tex].
4. Combine the results:
- Add the real parts: [tex]\(64 + (-25) = 39\)[/tex].
- The imaginary part remains [tex]\(-80i\)[/tex].
5. Write the final simplified product:
[tex]\[
39 - 80i
\][/tex]
Therefore, the correct product is [tex]\(39 - 80i\)[/tex].
1. Write the expression being multiplied:
[tex]\[
(8 - 5i) \times (8 - 5i)
\][/tex]
2. 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 term individually:
- [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], it becomes [tex]\(25 \times -1 = -25\)[/tex].
4. Combine the results:
- Add the real parts: [tex]\(64 + (-25) = 39\)[/tex].
- The imaginary part remains [tex]\(-80i\)[/tex].
5. Write the final simplified product:
[tex]\[
39 - 80i
\][/tex]
Therefore, the correct product is [tex]\(39 - 80i\)[/tex].
