Answer :
To solve the problem of multiplying and simplifying the expression [tex]\((8 - 5i)^2\)[/tex], we can use the formula for the square of a binomial. For a complex number in the form [tex]\((a + bi)\)[/tex], the square is given by:
[tex]\[
(a + bi)^2 = a^2 + 2abi + (bi)^2
\][/tex]
Here, the real part [tex]\( a = 8 \)[/tex] and the imaginary part [tex]\( b = -5 \)[/tex].
Now, let's go step-by-step:
1. Calculate [tex]\( a^2 \)[/tex] (the square of the real part):
[tex]\[
a^2 = 8^2 = 64
\][/tex]
2. Calculate [tex]\( 2ab \)[/tex] (the middle term involving both real and imaginary parts):
[tex]\[
2ab = 2 \cdot 8 \cdot (-5) = 2 \cdot -40 = -80
\][/tex]
3. Calculate [tex]\((bi)^2\)[/tex] (the square of the imaginary part):
[tex]\[
(bi)^2 = (-5i)^2 = (-5)^2 \cdot i^2 = 25 \cdot (-1) = -25
\][/tex]
Remember that [tex]\(i^2 = -1\)[/tex].
4. Combine the results to form the expression [tex]\((8 - 5i)^2\)[/tex]:
- Add the real parts: [tex]\(64 + (-25) = 39\)[/tex]
- The imaginary part is [tex]\(-80i\)[/tex].
Thus, the product is:
[tex]\[
39 - 80i
\][/tex]
So, the correct answer is [tex]\(89 - 80i\)[/tex].
[tex]\[
(a + bi)^2 = a^2 + 2abi + (bi)^2
\][/tex]
Here, the real part [tex]\( a = 8 \)[/tex] and the imaginary part [tex]\( b = -5 \)[/tex].
Now, let's go step-by-step:
1. Calculate [tex]\( a^2 \)[/tex] (the square of the real part):
[tex]\[
a^2 = 8^2 = 64
\][/tex]
2. Calculate [tex]\( 2ab \)[/tex] (the middle term involving both real and imaginary parts):
[tex]\[
2ab = 2 \cdot 8 \cdot (-5) = 2 \cdot -40 = -80
\][/tex]
3. Calculate [tex]\((bi)^2\)[/tex] (the square of the imaginary part):
[tex]\[
(bi)^2 = (-5i)^2 = (-5)^2 \cdot i^2 = 25 \cdot (-1) = -25
\][/tex]
Remember that [tex]\(i^2 = -1\)[/tex].
4. Combine the results to form the expression [tex]\((8 - 5i)^2\)[/tex]:
- Add the real parts: [tex]\(64 + (-25) = 39\)[/tex]
- The imaginary part is [tex]\(-80i\)[/tex].
Thus, the product is:
[tex]\[
39 - 80i
\][/tex]
So, the correct answer is [tex]\(89 - 80i\)[/tex].
