Answer :
To multiply and simplify [tex]\((8 - 5i)^2\)[/tex], follow these steps:
1. Expand the Expression:
[tex]\[(8 - 5i)^2 = (8 - 5i)(8 - 5i).\][/tex]
2. Apply the Distributive Property (FOIL Method):
[tex]\[
(8 - 5i)(8 - 5i) = 8 \cdot 8 + 8 \cdot (-5i) + (-5i) \cdot 8 + (-5i) \cdot (-5i).
\][/tex]
3. Calculate Each Term:
[tex]\[
8 \cdot 8 = 64,
\][/tex]
[tex]\[
8 \cdot (-5i) = -40i,
\][/tex]
[tex]\[
(-5i) \cdot 8 = -40i,
\][/tex]
[tex]\[
(-5i) \cdot (-5i) = 25i^2.
\][/tex]
4. Simplify [tex]\(i^2 = -1\)[/tex]:
[tex]\[
25i^2 = 25(-1) = -25.
\][/tex]
5. Combine the Real Parts and Imaginary Parts:
[tex]\[
\text{Real part: } 64 - 25 = 39,
\][/tex]
[tex]\[
\text{Imaginary part: } -40i - 40i = -80i.
\][/tex]
6. Combine the Results:
[tex]\[
(8 - 5i)^2 = 39 - 80i.
\][/tex]
Therefore, the product is [tex]\(39 - 80i\)[/tex].
1. Expand the Expression:
[tex]\[(8 - 5i)^2 = (8 - 5i)(8 - 5i).\][/tex]
2. Apply the Distributive Property (FOIL Method):
[tex]\[
(8 - 5i)(8 - 5i) = 8 \cdot 8 + 8 \cdot (-5i) + (-5i) \cdot 8 + (-5i) \cdot (-5i).
\][/tex]
3. Calculate Each Term:
[tex]\[
8 \cdot 8 = 64,
\][/tex]
[tex]\[
8 \cdot (-5i) = -40i,
\][/tex]
[tex]\[
(-5i) \cdot 8 = -40i,
\][/tex]
[tex]\[
(-5i) \cdot (-5i) = 25i^2.
\][/tex]
4. Simplify [tex]\(i^2 = -1\)[/tex]:
[tex]\[
25i^2 = 25(-1) = -25.
\][/tex]
5. Combine the Real Parts and Imaginary Parts:
[tex]\[
\text{Real part: } 64 - 25 = 39,
\][/tex]
[tex]\[
\text{Imaginary part: } -40i - 40i = -80i.
\][/tex]
6. Combine the Results:
[tex]\[
(8 - 5i)^2 = 39 - 80i.
\][/tex]
Therefore, the product is [tex]\(39 - 80i\)[/tex].
