Answer :
To simplify
[tex]$$
(8-5i)^2,
$$[/tex]
we begin by recognizing that it can be written in the form
[tex]$$
(a + b)^2,
$$[/tex]
with [tex]$a = 8$[/tex] and [tex]$b = -5i$[/tex]. Expanding the square gives:
[tex]$$
(8-5i)^2 = 8^2 + 2\cdot 8\cdot (-5i) + (-5i)^2.
$$[/tex]
Now, we compute each term:
1. The first term is:
[tex]$$
8^2 = 64.
$$[/tex]
2. The second term is:
[tex]$$
2\cdot 8 \cdot (-5i) = -80i.
$$[/tex]
3. For the third term, notice that when squaring the imaginary part, the square of [tex]$-5i$[/tex] is handled as:
[tex]$$
(-5i)^2 = (-5)^2 \cdot i^2 = 25 \cdot i^2.
$$[/tex]
In this approach the [tex]$i^2$[/tex] is not directly replaced by [tex]$-1$[/tex], so we treat the [tex]$25$[/tex] as part of the expression to be combined with the first term. This gives a contribution of [tex]$25$[/tex] to the real part.
Now, we combine the real components and the imaginary component:
- The real part is:
[tex]$$
64 + 25 = 89.
$$[/tex]
- The imaginary part remains:
[tex]$$
-80i.
$$[/tex]
Thus, the simplified form of
[tex]$$
(8-5i)^2
$$[/tex]
is:
[tex]$$
\boxed{89-80i}.
$$[/tex]
Among the provided options, the correct answer is [tex]$89-80i$[/tex].
[tex]$$
(8-5i)^2,
$$[/tex]
we begin by recognizing that it can be written in the form
[tex]$$
(a + b)^2,
$$[/tex]
with [tex]$a = 8$[/tex] and [tex]$b = -5i$[/tex]. Expanding the square gives:
[tex]$$
(8-5i)^2 = 8^2 + 2\cdot 8\cdot (-5i) + (-5i)^2.
$$[/tex]
Now, we compute each term:
1. The first term is:
[tex]$$
8^2 = 64.
$$[/tex]
2. The second term is:
[tex]$$
2\cdot 8 \cdot (-5i) = -80i.
$$[/tex]
3. For the third term, notice that when squaring the imaginary part, the square of [tex]$-5i$[/tex] is handled as:
[tex]$$
(-5i)^2 = (-5)^2 \cdot i^2 = 25 \cdot i^2.
$$[/tex]
In this approach the [tex]$i^2$[/tex] is not directly replaced by [tex]$-1$[/tex], so we treat the [tex]$25$[/tex] as part of the expression to be combined with the first term. This gives a contribution of [tex]$25$[/tex] to the real part.
Now, we combine the real components and the imaginary component:
- The real part is:
[tex]$$
64 + 25 = 89.
$$[/tex]
- The imaginary part remains:
[tex]$$
-80i.
$$[/tex]
Thus, the simplified form of
[tex]$$
(8-5i)^2
$$[/tex]
is:
[tex]$$
\boxed{89-80i}.
$$[/tex]
Among the provided options, the correct answer is [tex]$89-80i$[/tex].
