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 4i.

The simplified form is 2 + 2i.

The simplified form is 4 + 4i.

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 denominator by the conjugate of the denominator. The conjugate of \(1 - i\) is \(1 + i\).

Here's the step-by-step simplification:

1. Multiply the numerator and the denominator by \(1 + i\):

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

2. Simplifying the denominator:

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

3. Simultaneously, simplifying the numerator:

\[ (2 + 2i)(1 + i) = 2(1) + 2(i) + 2i(1) + 2i(i) = 2 + 2i + 2i + 2i^2 \] Since \(i^2 = -1\), we have:

\[ 2 + 2i + 2i - 2 = 0 + 4i = 4i \]

4. Putting it all together, we have:

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

Now we can assess the statements provided:

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

So, the true statements are:

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