To perform the operation and write the result in standard form, you don't necessarily have to multiply the top and bottom by i. However, doing so will simplify the expression and make it easier to work with.
Let's start by rewriting the expression:
(1 + i) / (i - 3) / (4 - i)
To simplify the expression, you can multiply the top and bottom by the conjugate of the denominator, which is (4 + i). This will eliminate the complex numbers in the denominator.
(1 + i) / (i - 3) * (4 + i) / (4 + i)
Next, we can use the distributive property to multiply the numerators and the denominators:
(1 + i)(4 + i) / (i - 3)(4 + i)
Now let's expand the numerator and denominator:
(4 + i + 4i + i^2) / (4i + i^2 - 12 - 3i)
Simplifying further:
(4 + 5i + i^2) / (i^2 + 4i - 12)
Since i^2 is equal to -1, we can substitute that in:
(4 + 5i - 1) / (-1 + 4i - 12)
Simplifying:
(3 + 5i) / (3 - 4i)
Now, to write the result in standard form, we need to rationalize the denominator. We can do this by multiplying both the numerator and the denominator by the conjugate of the denominator, which is (3 + 4i):
(3 + 5i) / (3 - 4i) * (3 + 4i) / (3 + 4i)
Multiplying the numerators and the denominators:
(9 + 12i + 15i + 20i^2) / (9 + 12i - 12i - 16i^2)
Simplifying further:
(9 + 27i - 20) / (9 + 16)
Simplifying:
(-11 + 27i) / 25
Thus, the result in standard form is -11/25 + (27/25)i.