Answer :
To find the product [tex]\((8-5i)^2\)[/tex], we'll use the formula for squaring a binomial, which is [tex]\((a + bi)^2 = a^2 + 2ab \cdot i + (bi)^2\)[/tex].
1. Identify the real and imaginary parts:
- The real part [tex]\(a\)[/tex] is 8.
- The imaginary part [tex]\(b\)[/tex] is [tex]\(-5\)[/tex].
2. Apply the formula:
- [tex]\(a^2 = 8^2 = 64\)[/tex]
- [tex]\(2ab \cdot i = 2 \cdot 8 \cdot (-5) \cdot i = -80i\)[/tex]
- [tex]\((bi)^2 = (-5i)^2 = 25 \cdot (-1) = -25\)[/tex] (Note: [tex]\(i^2 = -1\)[/tex])
3. Combine all parts:
[tex]\[
64 + (-80i) - 25
\][/tex]
4. Simplify:
- Combine the real parts: [tex]\(64 - 25 = 39\)[/tex]
- The imaginary part remains [tex]\(-80i\)[/tex].
So, the product [tex]\((8-5i)^2\)[/tex] simplifies to [tex]\(39 - 80i\)[/tex].
1. Identify the real and imaginary parts:
- The real part [tex]\(a\)[/tex] is 8.
- The imaginary part [tex]\(b\)[/tex] is [tex]\(-5\)[/tex].
2. Apply the formula:
- [tex]\(a^2 = 8^2 = 64\)[/tex]
- [tex]\(2ab \cdot i = 2 \cdot 8 \cdot (-5) \cdot i = -80i\)[/tex]
- [tex]\((bi)^2 = (-5i)^2 = 25 \cdot (-1) = -25\)[/tex] (Note: [tex]\(i^2 = -1\)[/tex])
3. Combine all parts:
[tex]\[
64 + (-80i) - 25
\][/tex]
4. Simplify:
- Combine the real parts: [tex]\(64 - 25 = 39\)[/tex]
- The imaginary part remains [tex]\(-80i\)[/tex].
So, the product [tex]\((8-5i)^2\)[/tex] simplifies to [tex]\(39 - 80i\)[/tex].
