Answer :
To multiply and simplify the product of the complex number [tex]\((8 - 5i)^2\)[/tex], we can use the formula for squaring a binomial:
[tex]\[
(a + bi)^2 = a^2 + 2abi + (bi)^2
\][/tex]
Here, [tex]\(a = 8\)[/tex] and [tex]\(b = -5\)[/tex]. Let’s calculate each part step by step:
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = 8^2 = 64
\][/tex]
2. Calculate [tex]\(2ab\)[/tex]:
[tex]\[
2ab = 2 \times 8 \times (-5) = -80i
\][/tex]
3. Calculate [tex]\((bi)^2\)[/tex]:
[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 results:
[tex]\[
a^2 + 2abi + (bi)^2 = 64 - 25 - 80i
\][/tex]
Simplify the real parts:
[tex]\[
64 - 25 = 39
\][/tex]
So, the simplified product is:
[tex]\[
39 - 80i
\][/tex]
Therefore, the product [tex]\((8 - 5i)^2\)[/tex] simplifies to [tex]\(39 - 80i\)[/tex].
[tex]\[
(a + bi)^2 = a^2 + 2abi + (bi)^2
\][/tex]
Here, [tex]\(a = 8\)[/tex] and [tex]\(b = -5\)[/tex]. Let’s calculate each part step by step:
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
a^2 = 8^2 = 64
\][/tex]
2. Calculate [tex]\(2ab\)[/tex]:
[tex]\[
2ab = 2 \times 8 \times (-5) = -80i
\][/tex]
3. Calculate [tex]\((bi)^2\)[/tex]:
[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 results:
[tex]\[
a^2 + 2abi + (bi)^2 = 64 - 25 - 80i
\][/tex]
Simplify the real parts:
[tex]\[
64 - 25 = 39
\][/tex]
So, the simplified product is:
[tex]\[
39 - 80i
\][/tex]
Therefore, the product [tex]\((8 - 5i)^2\)[/tex] simplifies to [tex]\(39 - 80i\)[/tex].
