To determine which of the given statements are true about the expression [tex]\((2 + 2i) + (1 - i)\)[/tex], we need to simplify and analyze this expression step-by-step.
### Step-by-Step Solution:
1. Identify the Real Parts and Imaginary Parts:
The given expression is:
[tex]\[
(2 + 2i) + (1 - i)
\][/tex]
- The real parts are: [tex]\(2\)[/tex] and [tex]\(1\)[/tex].
- The imaginary parts are: [tex]\(2i\)[/tex] and [tex]\(-i\)[/tex].
2. Sum the Real Parts:
Add the real parts together:
[tex]\[
2 + 1 = 3
\][/tex]
3. Sum the Imaginary Parts:
Add the imaginary parts together:
[tex]\[
2i - i = i
\][/tex]
4. Form the Simplified Expression:
Combine the summed real part and summed imaginary part:
[tex]\[
3 + i
\][/tex]
Therefore, the simplified form of the expression [tex]\((2 + 2i) + (1 - i)\)[/tex] is [tex]\(3 + i\)[/tex].
Now let's analyze the statements:
1. The simplified form is [tex]\(2i\)[/tex].
This is not true because the simplified form is [tex]\(3 + i\)[/tex].
2. The simplified form is [tex]\(4i\)[/tex].
This is not true because the simplified form is [tex]\(3 + i\)[/tex].
3. The simplified form is [tex]\(2 + 2i\)[/tex].
This is not true because the simplified form is [tex]\(3 + i\)[/tex].
4. The simplified form is [tex]\(4 + 4i\)[/tex].
This is not true because the simplified form is [tex]\(3 + i\)[/tex].
5. The simplified form is a complex number because complex numbers are closed under division.
This is true because the sum of [tex]\(3\)[/tex] (a real number) and [tex]\(i\)[/tex] (an imaginary number) is indeed a complex number, given by [tex]\(3 + i\)[/tex].
6. The simplified form is not a complex number because complex numbers are not closed under division.
This doesn't apply here because the simplified form we obtained [tex]\(3 + i\)[/tex] is indeed a complex number. Closure under division doesn't affect addition.
### Conclusion:
Based on the simplified form [tex]\(3 + i\)[/tex], the true statements are:
- The simplified form is [tex]\(3 + i\)[/tex] (corresponding to statement 5).
Therefore, the correct statements are:
[tex]\[ \boxed{5} \][/tex]
(Note: To match the answer provided, you mentioned the result as being [3, 5], but logically considering the provided statements, only statement 5 is true. The analysis above corrects any misconceptions from the given choices.)
