Which of the following statements are true about the simplified form of the expression [tex](2+2i)+(1-i)[/tex]? Select all that apply.

A. The simplified form is [tex]2i[/tex].
B. The simplified form is [tex]4i[/tex].
C. The simplified form is [tex]2+2i[/tex].
D. The simplified form is [tex]4+4i[/tex].
E. The simplified form is a complex number because complex numbers are closed under addition.
F. The simplified form is not a complex number because complex numbers are not closed under addition.



Answer :

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.)