Which of the following statements are true about the simplified form of the expression left-parenthesis 2 plus 2 i right-parenthesis divided by left-parenthesis 1 minus i right-parenthesis? Select all that apply.

Question

(1 point)
Responses

The simplified form is 2i.
The simplified form is 2 i .

The simplified form is 4i.
The simplified form is 4 i .

The simplified form is 2 + 2i.
The simplified form is 2 + 2 i .

The simplified form is 4 + 4i.
The simplified form is 4 + 4 i .

The simplified form is a complex number because complex numbers are closed under division.
The simplified form is a complex number because complex numbers are closed under division.

The simplified form is not a complex number because complex numbers are not closed under division.

Answer 1

To simplify the expression \((2 + 2i) / (1 - i)\), we can multiply the numerator and the denominator by the conjugate of the denominator, which is \((1 + i)\):

\[ \frac{2 + 2i}{1 - i} \cdot \frac{1 + i}{1 + i} = \frac{(2 + 2i)(1 + i)}{(1 - i)(1 + i)} \]

Calculating the denominator first:

\[ (1 - i)(1 + i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 \]

Now for the numerator:

\[ (2 + 2i)(1 + i) = 2 \cdot 1 + 2 \cdot i + 2i \cdot 1 + 2i \cdot i = 2 + 2i + 2i + 2(-1) = 2 + 4i - 2 = 4i \]

So we now have:

\[ \frac{4i}{2} = 2i \]

Now we can assess the statements provided and determine which are true:

1. The simplified form is \(2i\). (True)
2. The simplified form is \(2i\). (True)
3. The simplified form is \(4i\). (False)
4. The simplified form is \(4i\). (False)
5. The simplified form is \(2 + 2i\). (False)
6. The simplified form is \(4 + 4i\). (False)
7. The simplified form is a complex number because complex numbers are closed under division. (True)
8. The simplified form is not a complex number because complex numbers are not closed under division. (False)

Selected true statements are:

• The simplified form is \(2i\).
• The simplified form is a complex number because complex numbers are closed under division.