To find the value of the expression [tex]\((8 - 3i) - (8 - 3i)(8 + 8i)\)[/tex], let's break it down step-by-step.
1. Define the complex numbers:
Let [tex]\( a = 8 - 3i \)[/tex] and [tex]\( b = 8 + 8i \)[/tex].
2. Calculate the product of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
(8 - 3i)(8 + 8i)
\][/tex]
This can be expanded using the distributive property (FOIL method):
[tex]\[
(8 - 3i)(8 + 8i) = 8 \cdot 8 + 8 \cdot 8i - 3i \cdot 8 - 3i \cdot 8i
\][/tex]
Simplifying each term:
[tex]\[
= 64 + 64i - 24i - 24i^2
\][/tex]
Remember that [tex]\( i^2 = -1 \)[/tex], so [tex]\( -24i^2 = 24 \)[/tex]:
[tex]\[
= 64 + 64i - 24i + 24
\][/tex]
Combine the real and imaginary parts:
[tex]\[
= 88 + 40i
\][/tex]
3. Subtract the product from the first complex number [tex]\( a \)[/tex]:
[tex]\[
(8 - 3i) - (88 + 40i)
\][/tex]
Simplify by combining like terms (real parts and imaginary parts separately):
[tex]\[
= (8 - 88) + (-3i - 40i)
\][/tex]
[tex]\[
= -80 - 43i
\][/tex]
So, the value of the expression [tex]\((8 - 3i) - (8 - 3i)(8 + 8i)\)[/tex] is [tex]\(-80 - 43i\)[/tex].
Therefore, the correct answer is:
B. [tex]\(-80 - 43i\)[/tex]
