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Which expressions are equivalent to the given complex number [tex]45 + 2i[/tex]?

A. [tex](9 + 4i) + 2(4 + 7i)(1 - 2i)[/tex]
B. [tex](13 + 4i) + (32 - 6i)[/tex]
C. [tex](2 + 8i) + 2(9 + 6i)(1 - i)[/tex]
D. [tex](9 + 4i) + (36 - 2i)[/tex]
E. [tex](13 + 4i) + 2(7 + 2i)(2 - i)[/tex]
F. [tex](2 + 8i) + (30 - 6i)[/tex]

Sagot :

GinnyAnswer
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]
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