Answer :
To solve the problem [tex]\((8-5i)^2\)[/tex], we need to multiply the expression out and simplify it. Here's a step-by-step breakdown of how to do this:
1. Identify the components: We have [tex]\((8 - 5i)\)[/tex] that we need to square, which gives us [tex]\((8 - 5i) \times (8 - 5i)\)[/tex].
2. Apply the formula: Use the formula [tex]\((a - bi)^2 = a^2 - 2abi + (bi)^2\)[/tex] to expand the expression. In our case, [tex]\(a = 8\)[/tex] and [tex]\(b = 5\)[/tex].
3. Calculate each term:
- First term: [tex]\(a^2 = 8^2 = 64\)[/tex]
- Middle term: [tex]\(-2abi = -2 \times 8 \times 5i = -80i\)[/tex]
- Last term: [tex]\((bi)^2 = (5i)^2 = 25i^2\)[/tex]. Since [tex]\(i^2 = -1\)[/tex], this becomes [tex]\(25 \times -1 = -25\)[/tex].
4. Combine the terms:
Combine the real parts and the imaginary parts separately:
- Real part: [tex]\(64 - 25 = 39\)[/tex]
- Imaginary part: [tex]\(-80i\)[/tex]
5. Final simplified result:
The expression [tex]\((8 - 5i)^2\)[/tex] simplifies to [tex]\(39 - 80i\)[/tex].
Therefore, the product is 39 - 80i.
1. Identify the components: We have [tex]\((8 - 5i)\)[/tex] that we need to square, which gives us [tex]\((8 - 5i) \times (8 - 5i)\)[/tex].
2. Apply the formula: Use the formula [tex]\((a - bi)^2 = a^2 - 2abi + (bi)^2\)[/tex] to expand the expression. In our case, [tex]\(a = 8\)[/tex] and [tex]\(b = 5\)[/tex].
3. Calculate each term:
- First term: [tex]\(a^2 = 8^2 = 64\)[/tex]
- Middle term: [tex]\(-2abi = -2 \times 8 \times 5i = -80i\)[/tex]
- Last term: [tex]\((bi)^2 = (5i)^2 = 25i^2\)[/tex]. Since [tex]\(i^2 = -1\)[/tex], this becomes [tex]\(25 \times -1 = -25\)[/tex].
4. Combine the terms:
Combine the real parts and the imaginary parts separately:
- Real part: [tex]\(64 - 25 = 39\)[/tex]
- Imaginary part: [tex]\(-80i\)[/tex]
5. Final simplified result:
The expression [tex]\((8 - 5i)^2\)[/tex] simplifies to [tex]\(39 - 80i\)[/tex].
Therefore, the product is 39 - 80i.
