To simplify the expression \((2 + 2i) / (1 - i)\), we can multiply the numerator and the denominator by the conjugate of the denominator, which is \(1 + i\).
So, we have:
\[ \frac{(2 + 2i)(1 + i)}{(1 - i)(1 + i)} \]
Calculating the denominator:
\[ (1 - i)(1 + i) = 1^2 - i^2 = 1 - (-1) = 2 \]
Now, calculating the numerator:
\[ (2 + 2i)(1 + i) = 2(1) + 2(i) + 2i(1) + 2i(i) = 2 + 2i + 2i - 2 = 2 + 4i - 2 = 4i \]
So, combining the results, we have:
\[ \frac{4i}{2} = 2i \]
Now let's analyze the statements:
- The simplified form is 2i - True (This is what we found)
- The simplified form is 4i - False (We found it to be 2i)
- The simplified form is 2 + 2i - False (We found it to be 2i)
- The simplified form is 4 + 4i - False (We found it to be 2i)
- The simplified form is a complex number because complex numbers are closed under division. - True (The result is indeed a complex number)
- The simplified form is not a complex number because complex numbers are not closed under division. - False (That statement is incorrect; complex numbers are closed under division, except when dividing by zero)
In conclusion, the true statements are:
- The simplified form is 2i.
- The simplified form is a complex number because complex numbers are closed under division.