Answer :
To solve the problem [tex]\((8 - 5i)^2\)[/tex] and simplify the product, follow these steps:
1. Write down the formula for the square of a complex number:
[tex]\[
(a - bi)^2 = a^2 - 2abi + (bi)^2
\][/tex]
2. Identify the real and imaginary parts:
In the complex number [tex]\(8 - 5i\)[/tex], the real part [tex]\(a\)[/tex] is 8, and the imaginary part [tex]\(b\)[/tex] is 5.
3. Square the real part:
[tex]\[
a^2 = 8^2 = 64
\][/tex]
4. Square the imaginary part and include the negative sign of [tex]\(i^2 = -1\)[/tex]:
[tex]\[
(bi)^2 = (-5i)^2 = 25i^2 = 25(-1) = -25
\][/tex]
5. Calculate the middle term [tex]\(-2ab\)[/tex]:
[tex]\[
-2abi = -2(8)(5)i = -80i
\][/tex]
6. Combine all the parts:
[tex]\[
(8 - 5i)^2 = a^2 - 2abi + (bi)^2 = 64 - 80i - 25
\][/tex]
7. Combine the real components:
[tex]\[
64 - 25 = 39
\][/tex]
So, [tex]\((8 - 5i)^2\)[/tex] simplifies to:
[tex]\[
39 - 80i
\][/tex]
Thus, the correct product is [tex]\(39 - 80i\)[/tex].
1. Write down the formula for the square of a complex number:
[tex]\[
(a - bi)^2 = a^2 - 2abi + (bi)^2
\][/tex]
2. Identify the real and imaginary parts:
In the complex number [tex]\(8 - 5i\)[/tex], the real part [tex]\(a\)[/tex] is 8, and the imaginary part [tex]\(b\)[/tex] is 5.
3. Square the real part:
[tex]\[
a^2 = 8^2 = 64
\][/tex]
4. Square the imaginary part and include the negative sign of [tex]\(i^2 = -1\)[/tex]:
[tex]\[
(bi)^2 = (-5i)^2 = 25i^2 = 25(-1) = -25
\][/tex]
5. Calculate the middle term [tex]\(-2ab\)[/tex]:
[tex]\[
-2abi = -2(8)(5)i = -80i
\][/tex]
6. Combine all the parts:
[tex]\[
(8 - 5i)^2 = a^2 - 2abi + (bi)^2 = 64 - 80i - 25
\][/tex]
7. Combine the real components:
[tex]\[
64 - 25 = 39
\][/tex]
So, [tex]\((8 - 5i)^2\)[/tex] simplifies to:
[tex]\[
39 - 80i
\][/tex]
Thus, the correct product is [tex]\(39 - 80i\)[/tex].
