Answer :
Sure! Let's solve the problem step-by-step.
We need to multiply and simplify the product of [tex]\((8 - 5i)^2\)[/tex].
1. Write out the expression:
[tex]\[
(8 - 5i)^2
\][/tex]
2. Rewrite the expression using binomial expansion:
[tex]\[
(8 - 5i) \times (8 - 5i)
\][/tex]
3. Apply the distributive property (FOIL method):
[tex]\[
(8 - 5i)(8 - 5i) = 8 \times 8 + 8 \times (-5i) + (-5i) \times 8 + (-5i) \times (-5i)
\][/tex]
4. Perform the multiplications:
[tex]\[
= 64 - 40i - 40i + 25i^2
\][/tex]
5. Simplify [tex]\(i^2\)[/tex] to [tex]\(-1\)[/tex]:
[tex]\[
= 64 - 40i - 40i + 25(-1)
\][/tex]
6. Combine like terms:
[tex]\[
= 64 - 80i - 25
\][/tex]
7. Combine the real parts and the imaginary parts:
[tex]\[
= (64 - 25) - 80i
\][/tex]
[tex]\[
= 39 - 80i
\][/tex]
Thus, the product [tex]\((8 - 5i)^2\)[/tex] simplifies to [tex]\(\boxed{39 - 80i}\)[/tex].
We need to multiply and simplify the product of [tex]\((8 - 5i)^2\)[/tex].
1. Write out the expression:
[tex]\[
(8 - 5i)^2
\][/tex]
2. Rewrite the expression using binomial expansion:
[tex]\[
(8 - 5i) \times (8 - 5i)
\][/tex]
3. Apply the distributive property (FOIL method):
[tex]\[
(8 - 5i)(8 - 5i) = 8 \times 8 + 8 \times (-5i) + (-5i) \times 8 + (-5i) \times (-5i)
\][/tex]
4. Perform the multiplications:
[tex]\[
= 64 - 40i - 40i + 25i^2
\][/tex]
5. Simplify [tex]\(i^2\)[/tex] to [tex]\(-1\)[/tex]:
[tex]\[
= 64 - 40i - 40i + 25(-1)
\][/tex]
6. Combine like terms:
[tex]\[
= 64 - 80i - 25
\][/tex]
7. Combine the real parts and the imaginary parts:
[tex]\[
= (64 - 25) - 80i
\][/tex]
[tex]\[
= 39 - 80i
\][/tex]
Thus, the product [tex]\((8 - 5i)^2\)[/tex] simplifies to [tex]\(\boxed{39 - 80i}\)[/tex].
