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]
