Answer :
To simplify the expression
[tex]$$
(8-5i)^2,
$$[/tex]
we start by writing the square as a product:
[tex]$$
(8-5i)(8-5i).
$$[/tex]
Now, we expand the product using the distributive property (FOIL method):
1. Multiply the first terms:
[tex]$$
8 \times 8 = 64.
$$[/tex]
2. Multiply the outer terms:
[tex]$$
8 \times (-5i) = -40i.
$$[/tex]
3. Multiply the inner terms:
[tex]$$
(-5i) \times 8 = -40i.
$$[/tex]
4. Multiply the last terms:
[tex]$$
(-5i) \times (-5i) = 25i^2.
$$[/tex]
Next, add all these results together:
[tex]$$
64 - 40i - 40i + 25i^2 = 64 - 80i + 25i^2.
$$[/tex]
Recall that [tex]$i^2 = -1$[/tex]. Thus, we substitute:
[tex]$$
25i^2 = 25(-1) = -25.
$$[/tex]
Now, replace [tex]$25i^2$[/tex] with [tex]$-25$[/tex] in the expression:
[tex]$$
64 - 80i - 25.
$$[/tex]
Finally, combine the real numbers:
[tex]$$
64 - 25 = 39.
$$[/tex]
So, the simplified form is:
[tex]$$
39 - 80i.
$$[/tex]
Thus, the product is [tex]$\boxed{39-80i}$[/tex].
[tex]$$
(8-5i)^2,
$$[/tex]
we start by writing the square as a product:
[tex]$$
(8-5i)(8-5i).
$$[/tex]
Now, we expand the product using the distributive property (FOIL method):
1. Multiply the first terms:
[tex]$$
8 \times 8 = 64.
$$[/tex]
2. Multiply the outer terms:
[tex]$$
8 \times (-5i) = -40i.
$$[/tex]
3. Multiply the inner terms:
[tex]$$
(-5i) \times 8 = -40i.
$$[/tex]
4. Multiply the last terms:
[tex]$$
(-5i) \times (-5i) = 25i^2.
$$[/tex]
Next, add all these results together:
[tex]$$
64 - 40i - 40i + 25i^2 = 64 - 80i + 25i^2.
$$[/tex]
Recall that [tex]$i^2 = -1$[/tex]. Thus, we substitute:
[tex]$$
25i^2 = 25(-1) = -25.
$$[/tex]
Now, replace [tex]$25i^2$[/tex] with [tex]$-25$[/tex] in the expression:
[tex]$$
64 - 80i - 25.
$$[/tex]
Finally, combine the real numbers:
[tex]$$
64 - 25 = 39.
$$[/tex]
So, the simplified form is:
[tex]$$
39 - 80i.
$$[/tex]
Thus, the product is [tex]$\boxed{39-80i}$[/tex].
