Solve: (4 - rad 3 i)+3
/(4-rad 3 i)-3
To solve the expression ((4 - sqrt(3)i) + 3) / ((4 - sqrt(3)i) - 3), we can simplify the expression in the numerator and the expression in the denominator separately and then divide the results.
Step 1: Simplify the expression in the numerator.
To simplify (4 - sqrt(3)i) + 3, we add the real parts separately and the imaginary parts separately:
Real parts: 4 + 3 = 7
Imaginary parts: -sqrt(3)i + 0i = -sqrt(3)i
So, the numerator simplifies to 7 - sqrt(3)i.
Step 2: Simplify the expression in the denominator.
To simplify (4 - sqrt(3)i) - 3, we subtract the real parts separately and the imaginary parts separately:
Real parts: 4 - 3 = 1
Imaginary parts: -sqrt(3)i - 0i = -sqrt(3)i
So, the denominator simplifies to 1 - sqrt(3)i.
Step 3: Divide the numerator by the denominator.
Dividing (7 - sqrt(3)i) by (1 - sqrt(3)i) requires multiplying both the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator.
The conjugate of (1 - sqrt(3)i) is (1 + sqrt(3)i). Therefore, we can multiply both the numerator and denominator by (1 + sqrt(3)i):
((7 - sqrt(3)i) * (1 + sqrt(3)i)) / ((1 - sqrt(3)i) * (1 + sqrt(3)i))
Expanding the numerator and denominator, we get:
((7 - sqrt(3)i + 7sqrt(3)i - 3) / (1 - (sqrt(3)i)^2))
Simplifying further:
(4 + 6sqrt(3)i) / (1 - (-3))
(4 + 6sqrt(3)i) / 4
Canceling out the common factor of 2 from the numerator and denominator, we finally get:
2 + 3sqrt(3)i as the simplified form of the expression.
Therefore, ((4 - sqrt(3)i) + 3) / ((4 - sqrt(3)i) - 3) = 2 + 3sqrt(3)i.