Answer :
To find the product and simplify [tex]\((8 - 5i)^2\)[/tex], we will follow these steps:
1. Understand the Formula: We need to square the complex number [tex]\((8 - 5i)\)[/tex]. The formula for the square of a complex number [tex]\( (a + bi)^2 \)[/tex] is:
[tex]\[
(a + bi)^2 = a^2 + 2abi + (bi)^2
\][/tex]
where [tex]\( i^2 = -1 \)[/tex].
2. Apply the Values: For our complex number [tex]\( 8 - 5i \)[/tex]:
[tex]\[
(8 - 5i)^2
\][/tex]
3. Expand Using the Formula:
[tex]\[
(8 - 5i)^2 = (8)^2 + 2 \cdot 8 \cdot (-5i) + (-5i)^2
\][/tex]
4. Calculate Each Part:
- The square of the real part [tex]\( a = 8 \)[/tex]:
[tex]\[
8^2 = 64
\][/tex]
- The product [tex]\( 2 \cdot 8 \cdot (-5i) \)[/tex]:
[tex]\[
2 \cdot 8 \cdot (-5i) = 16 \cdot (-5i) = -80i
\][/tex]
- The square of the imaginary part [tex]\( b = -5i \)[/tex]:
[tex]\[
(-5i)^2 = (-5)^2 \cdot i^2 = 25 \cdot (-1) = -25
\][/tex]
5. Combine Real and Imaginary Parts:
[tex]\[
64 + (-25) + (-80i) = (64 - 25) + (-80i) = 39 - 80i
\][/tex]
So, the product and simplified form of [tex]\((8 - 5i)^2\)[/tex] is:
[tex]\[
\boxed{39 - 80i}
\][/tex]
1. Understand the Formula: We need to square the complex number [tex]\((8 - 5i)\)[/tex]. The formula for the square of a complex number [tex]\( (a + bi)^2 \)[/tex] is:
[tex]\[
(a + bi)^2 = a^2 + 2abi + (bi)^2
\][/tex]
where [tex]\( i^2 = -1 \)[/tex].
2. Apply the Values: For our complex number [tex]\( 8 - 5i \)[/tex]:
[tex]\[
(8 - 5i)^2
\][/tex]
3. Expand Using the Formula:
[tex]\[
(8 - 5i)^2 = (8)^2 + 2 \cdot 8 \cdot (-5i) + (-5i)^2
\][/tex]
4. Calculate Each Part:
- The square of the real part [tex]\( a = 8 \)[/tex]:
[tex]\[
8^2 = 64
\][/tex]
- The product [tex]\( 2 \cdot 8 \cdot (-5i) \)[/tex]:
[tex]\[
2 \cdot 8 \cdot (-5i) = 16 \cdot (-5i) = -80i
\][/tex]
- The square of the imaginary part [tex]\( b = -5i \)[/tex]:
[tex]\[
(-5i)^2 = (-5)^2 \cdot i^2 = 25 \cdot (-1) = -25
\][/tex]
5. Combine Real and Imaginary Parts:
[tex]\[
64 + (-25) + (-80i) = (64 - 25) + (-80i) = 39 - 80i
\][/tex]
So, the product and simplified form of [tex]\((8 - 5i)^2\)[/tex] is:
[tex]\[
\boxed{39 - 80i}
\][/tex]
