Answer :
To multiply and simplify the product [tex]\((8 - 5i)^2\)[/tex], we need to use the formula for squaring a binomial, which is [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].
Let's break this down step by step:
1. Identify the components:
- Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
2. Calculate [tex]\(a^2\)[/tex]:
- [tex]\(a^2 = 8^2 = 64\)[/tex].
3. Calculate [tex]\(-2ab\)[/tex]:
- First, calculate [tex]\(ab = 8 \times 5i = 40i\)[/tex].
- Then, [tex]\(-2ab = -2 \times 40i = -80i\)[/tex].
4. Calculate [tex]\(b^2\)[/tex]:
- Since [tex]\(b = 5i\)[/tex], we have [tex]\(b^2 = (5i)^2 = 25i^2\)[/tex].
- We know that [tex]\(i^2 = -1\)[/tex], so [tex]\(25i^2 = 25 \times -1 = -25\)[/tex].
5. Combine the results:
- Now add all these together: [tex]\(a^2 + (-2ab) + b^2 = 64 - 80i - 25\)[/tex].
6. Simplify the expression:
- Combine the real parts: [tex]\(64 - 25 = 39\)[/tex].
- The imaginary part remains [tex]\(-80i\)[/tex].
Putting it all together, the simplified product is:
[tex]\[39 - 80i\][/tex]
Thus, the correct answer is [tex]\(39 - 80i\)[/tex].
Let's break this down step by step:
1. Identify the components:
- Here, [tex]\(a = 8\)[/tex] and [tex]\(b = 5i\)[/tex].
2. Calculate [tex]\(a^2\)[/tex]:
- [tex]\(a^2 = 8^2 = 64\)[/tex].
3. Calculate [tex]\(-2ab\)[/tex]:
- First, calculate [tex]\(ab = 8 \times 5i = 40i\)[/tex].
- Then, [tex]\(-2ab = -2 \times 40i = -80i\)[/tex].
4. Calculate [tex]\(b^2\)[/tex]:
- Since [tex]\(b = 5i\)[/tex], we have [tex]\(b^2 = (5i)^2 = 25i^2\)[/tex].
- We know that [tex]\(i^2 = -1\)[/tex], so [tex]\(25i^2 = 25 \times -1 = -25\)[/tex].
5. Combine the results:
- Now add all these together: [tex]\(a^2 + (-2ab) + b^2 = 64 - 80i - 25\)[/tex].
6. Simplify the expression:
- Combine the real parts: [tex]\(64 - 25 = 39\)[/tex].
- The imaginary part remains [tex]\(-80i\)[/tex].
Putting it all together, the simplified product is:
[tex]\[39 - 80i\][/tex]
Thus, the correct answer is [tex]\(39 - 80i\)[/tex].
