Answer :
To solve the problem [tex]\((8 - 5i)^2\)[/tex], let's follow these steps:
1. Apply the formula for squaring a binomial:
[tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]
2. Substitute the terms:
Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
3. Calculate each part:
- [tex]\(a^2 = 8^2 = 64\)[/tex]
- [tex]\(b^2 = (5i)^2 = 25i^2\)[/tex]. Since [tex]\(i^2 = -1\)[/tex], this becomes [tex]\(25 \times -1 = -25\)[/tex].
- The middle term is [tex]\(-2ab = -2 \times 8 \times 5i = -80i\)[/tex].
4. Combine the results:
- The real part: [tex]\(a^2 + b^2 = 64 - 25 = 39\)[/tex]
- The imaginary part: [tex]\(-80i\)[/tex]
5. Final result:
Combine the real and imaginary parts to get [tex]\(39 - 80i\)[/tex].
Therefore, the simplified form of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex]. The correct option is [tex]\(39 - 80i\)[/tex].
1. Apply the formula for squaring a binomial:
[tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex]
2. Substitute the terms:
Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
3. Calculate each part:
- [tex]\(a^2 = 8^2 = 64\)[/tex]
- [tex]\(b^2 = (5i)^2 = 25i^2\)[/tex]. Since [tex]\(i^2 = -1\)[/tex], this becomes [tex]\(25 \times -1 = -25\)[/tex].
- The middle term is [tex]\(-2ab = -2 \times 8 \times 5i = -80i\)[/tex].
4. Combine the results:
- The real part: [tex]\(a^2 + b^2 = 64 - 25 = 39\)[/tex]
- The imaginary part: [tex]\(-80i\)[/tex]
5. Final result:
Combine the real and imaginary parts to get [tex]\(39 - 80i\)[/tex].
Therefore, the simplified form of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(39 - 80i\)[/tex]. The correct option is [tex]\(39 - 80i\)[/tex].
