To determine which of the given expressions are equivalent to the complex number [tex]$45 + 2i$[/tex], we need to evaluate each expression step-by-step and compare the results.
1. Expression 1:
[tex]\[
(9 + 4i) + 2(4 + 7i)(1 - 2i)
\][/tex]
Evaluating this, we find:
[tex]\[
(45 + 2i)
\][/tex]
This is equivalent to the given complex number [tex]$45 + 2i$[/tex].
2. Expression 2:
[tex]\[
(13 + 4i) + (32 - 6i)
\][/tex]
Evaluating this, we find:
[tex]\[
(45 - 2i)
\][/tex]
This is not equivalent to the given complex number [tex]$45 + 2i$[/tex].
3. Expression 3:
[tex]\[
(2 + 8i) + 2(9 + 6i)(1 - i)
\][/tex]
Evaluating this, we find:
[tex]\[
(32 + 2i)
\][/tex]
This is not equivalent to the given complex number [tex]$45 + 2i$[/tex].
4. Expression 4:
[tex]\[
(9 + 4i) + (36 - 2i)
\][/tex]
Evaluating this, we find:
[tex]\[
(45 + 2i)
\][/tex]
This is equivalent to the given complex number [tex]$45 + 2i$[/tex].
5. Expression 5:
[tex]\[
(13 + 4i) + 2(7 + 2i)(2 - i)
\][/tex]
Evaluating this, we find:
[tex]\[
(45 - 2i)
\][/tex]
This is not equivalent to the given complex number [tex]$45 + 2i$[/tex].
6. Expression 6:
[tex]\[
(2 + 8i) + (30 - 6i)
\][/tex]
Evaluating this, we find:
[tex]\[
(32 + 2i)
\][/tex]
This is not equivalent to the given complex number [tex]$45 + 2i$[/tex].
Thus, the expressions that are equivalent to [tex]$45 + 2i$[/tex] are:
[tex]\[
(9 + 4i) + 2(4 + 7i)(1 - 2i)
\][/tex]
and
[tex]\[
(9 + 4i) + (36 - 2i)
\][/tex]
