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Which complex number is equivalent to this expression?

[tex]\[7 + (9 + 27i) - 2(25 - 11i)\][/tex]

A. [tex]\[66 - 49i\][/tex]

B. [tex]\[-34 + 5i\][/tex]

C. [tex]\[68 + 5i\][/tex]

D. [tex]\[-34 + 49i\][/tex]

Sagot :

GinnyAnswer
To solve the given complex number expression step by step, let's break it down and simplify it systematically.

Given expression:
[tex]\[ 7 + (9 + 27i) - 2(25 - 11i) \][/tex]

First, apply the distributive property to [tex]\( -2(25 - 11i) \)[/tex]:
[tex]\[ -2 \times 25 = -50 \][/tex]
[tex]\[ -2 \times (-11i) = 22i \][/tex]
So, the expression [tex]\( -2(25 - 11i) \)[/tex] simplifies to:
[tex]\[ -50 + 22i \][/tex]

Now rewrite the entire expression with this simplified form:
[tex]\[ 7 + (9 + 27i) - 50 + 22i \][/tex]

Combine the real parts and the imaginary parts separately:
1. Real parts: [tex]\( 7 + 9 - 50 \)[/tex]
[tex]\[ 7 + 9 = 16 \][/tex]
[tex]\[ 16 - 50 = -34 \][/tex]

2. Imaginary parts: [tex]\( 27i + 22i \)[/tex]
[tex]\[ 27i + 22i = 49i \][/tex]

Thus, the combined simplified form is:
[tex]\[ -34 + 49i \][/tex]

Hence, the equivalent complex number is:
[tex]\[ \boxed{-34 + 49i} \][/tex]

So, the correct answer is:
D. [tex]\(-34 + 49i\)[/tex]
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