Answer :
To multiply and simplify the product [tex]\((8 - 5i)^2\)[/tex], you can use the formula for squaring a binomial, which is [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].
1. Identify the values in the expression:
- [tex]\(a = 8\)[/tex]
- [tex]\(b = 5\)[/tex]
2. Apply the formula:
- Calculate [tex]\(a^2\)[/tex]:
[tex]\[
8^2 = 64
\][/tex]
- Calculate [tex]\(-2ab\)[/tex]:
[tex]\[
-2 \times 8 \times 5 = -80i
\][/tex]
- Calculate [tex]\(b^2\)[/tex]:
[tex]\[
(5i)^2 = 25i^2 = 25 \times (-1) = -25
\][/tex]
(Remember that [tex]\(i^2 = -1\)[/tex])
3. Combine the results:
- The real part is [tex]\(a^2 + b^2\)[/tex]:
[tex]\[
64 - 25 = 39
\][/tex]
- The imaginary part is [tex]\(-2ab\)[/tex]:
[tex]\[
-80i
\][/tex]
So, the expression [tex]\((8 - 5i)^2\)[/tex] simplifies to [tex]\(39 - 80i\)[/tex].
Therefore, the correct product is 39 - 80i.
1. Identify the values in the expression:
- [tex]\(a = 8\)[/tex]
- [tex]\(b = 5\)[/tex]
2. Apply the formula:
- Calculate [tex]\(a^2\)[/tex]:
[tex]\[
8^2 = 64
\][/tex]
- Calculate [tex]\(-2ab\)[/tex]:
[tex]\[
-2 \times 8 \times 5 = -80i
\][/tex]
- Calculate [tex]\(b^2\)[/tex]:
[tex]\[
(5i)^2 = 25i^2 = 25 \times (-1) = -25
\][/tex]
(Remember that [tex]\(i^2 = -1\)[/tex])
3. Combine the results:
- The real part is [tex]\(a^2 + b^2\)[/tex]:
[tex]\[
64 - 25 = 39
\][/tex]
- The imaginary part is [tex]\(-2ab\)[/tex]:
[tex]\[
-80i
\][/tex]
So, the expression [tex]\((8 - 5i)^2\)[/tex] simplifies to [tex]\(39 - 80i\)[/tex].
Therefore, the correct product is 39 - 80i.
