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What is the value of the expression given below?

[tex](8-3i)-(8-3i)(8+8i)[/tex]

A. [tex]-80 + 494i[/tex]
B. [tex]-80 - 431i[/tex]
C. [tex]-96 - 373i[/tex]
D. [tex]-88 + 372i[/tex]


Sagot :

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]