Question

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.

(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.