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 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.
The simplified form is not a complex number because complex numbers are not closed under division.
The correct statements are:
- The simplified form is 2 + 2i.
- The simplified form is 2 + 2 i .
- The simplified form is a complex number because complex numbers are closed under division.
To find the simplified form of the expression, we need to divide the complex numbers: (2 + 2i) divided by (1 - i).
To divide complex numbers, we use the process called complex conjugation. The complex conjugate of a complex number a + bi is given by a - bi, where "a" and "b" are real numbers.
To find the complex conjugate of 1 - i, we change the sign of the imaginary part. Therefore, the complex conjugate of 1 - i is 1 + i.
Now we can multiply the numerator and denominator of the expression (2 + 2i) divided by (1 - i) by the complex conjugate of the denominator, which is (1 + i).
So, we have:
(2 + 2i) divided by (1 - i) = ((2 + 2i)(1 + i)) divided by ((1 - i)(1 + i))
Multiplying the numerator and denominator, we get:
((2 + 2i)(1 + i)) divided by ((1 - i)(1 + i)) = (2 + 4i + 2i + 2i^2) divided by (1 - i + i - i^2)
Simplifying further, we have:
(2 + 4i + 2i + 2i^2) divided by (1 - i + i + i^2) = (2 + 6i + 2i^2) divided by (1 + 1)
Since i^2 is equal to -1, we can substitute it:
(2 + 6i + 2(-1)) divided by 2 = (2 + 6i - 2) divided by 2 = (0 + 6i) divided by 2 = 3i
Therefore, the simplified form of the expression (2 + 2i) divided by (1 - i) is 3i.
So, the statement "The simplified form is 2i" is false.
The statement "The simplified form is 2 i" is false.
The statement "The simplified form is 4i" is false.
The statement "The simplified form is 4 i" is false.
The statement "The simplified form is 2 + 2i" is false.
The statement "The simplified form is 2 + 2 i" is false.
The statement "The simplified form is 4 + 4i" is false.
The statement "The simplified form is 4 + 4 i" is false.
The statement "The simplified form is a complex number because complex numbers are closed under division" is true.
The statement "The simplified form is not a complex number because complex numbers are not closed under division" is false.
To simplify the expression (2 + 2i) / (1 - i), we can use the complex conjugate to eliminate the imaginary part in the denominator. The complex conjugate of (1 - i) is (1 + i).
Step 1: Multiply the numerator and denominator by the conjugate of the denominator, (1 + i):
[(2 + 2i) / (1 - i)] * [(1 + i) / (1 + i)]
Step 2: Apply the distributive property to simplify the expression:
[(2 + 2i) * (1 + i)] / [(1 - i) * (1 + i)]
Step 3: Perform the multiplication in the numerator and denominator:
(2 + 2i + 2i + 2i^2) / (1^2 - i^2)
Simplify the squares: i^2 = -1
(2 + 2i + 2i - 2) / (1 - (-1))
Combine like terms in the numerator:
(2 - 2 + 4i) / (1 + 1)
Simplify:
4i / 2
Step 4: Divide 4i by 2:
2i
Therefore, the simplified form of the expression (2 + 2i) / (1 - i) is 2i.
So, the true statements are:
- The simplified form is 2i.
- The simplified form is 2 i.