Answer :
Sure! Let's solve the problem step by step.
We need to multiply and simplify the expression [tex]\((8 - 5i)^2\)[/tex].
### Step 1: Recognize the Binomial Square Formula
The expression [tex]\((8 - 5i)^2\)[/tex] is in the form of [tex]\((a - b)^2\)[/tex], which can be expanded using the formula:
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]
### Step 2: Apply the Formula
In the expression [tex]\((8 - 5i)\)[/tex], we identify:
- [tex]\(a = 8\)[/tex]
- [tex]\(b = 5i\)[/tex]
Using the formula, we have:
[tex]\[
a^2 = 8^2 = 64
\][/tex]
[tex]\[
b^2 = (5i)^2 = 25i^2
\][/tex]
Remember that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
b^2 = 25(-1) = -25
\][/tex]
[tex]\[
2ab = 2(8)(5i) = 80i
\][/tex]
### Step 3: Substitute and Simplify
Substituting back into the formula gives us:
[tex]\[
a^2 - 2ab + b^2 = 64 - 80i - 25
\][/tex]
Now, combine the real parts:
[tex]\[
64 - 25 = 39
\][/tex]
Thus, the expression simplifies to:
[tex]\[
39 - 80i
\][/tex]
So, the product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(\boxed{39 - 80i}\)[/tex].
We need to multiply and simplify the expression [tex]\((8 - 5i)^2\)[/tex].
### Step 1: Recognize the Binomial Square Formula
The expression [tex]\((8 - 5i)^2\)[/tex] is in the form of [tex]\((a - b)^2\)[/tex], which can be expanded using the formula:
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]
### Step 2: Apply the Formula
In the expression [tex]\((8 - 5i)\)[/tex], we identify:
- [tex]\(a = 8\)[/tex]
- [tex]\(b = 5i\)[/tex]
Using the formula, we have:
[tex]\[
a^2 = 8^2 = 64
\][/tex]
[tex]\[
b^2 = (5i)^2 = 25i^2
\][/tex]
Remember that [tex]\(i^2 = -1\)[/tex], so:
[tex]\[
b^2 = 25(-1) = -25
\][/tex]
[tex]\[
2ab = 2(8)(5i) = 80i
\][/tex]
### Step 3: Substitute and Simplify
Substituting back into the formula gives us:
[tex]\[
a^2 - 2ab + b^2 = 64 - 80i - 25
\][/tex]
Now, combine the real parts:
[tex]\[
64 - 25 = 39
\][/tex]
Thus, the expression simplifies to:
[tex]\[
39 - 80i
\][/tex]
So, the product of [tex]\((8 - 5i)^2\)[/tex] is [tex]\(\boxed{39 - 80i}\)[/tex].
