Question

Perform the operation and write the result in standard form.

1 + i over i - 3 over 4 - i

Do I have to multiply top and bottom by i?

Answer 1

For
(1+i)/(i-3)
you multiply top and bottom by the conjugate of the denominator, i.e. for i-3, it will be i+3.
Thus
(1+i)/(i-3)
=(i+3)(1+i) / ((i+3)(i-3))
=(i²+4i+3)/(i²+3i-3i-9)
=(-1+4i+3)/(-1-9)
=(4i+2)/(-10)
=-2(2i+1)/(2*5)
=-(2i+1)/5

Try using this technique to apply to your problem.

Answer 2

Why conjugage?

The conjugage of a given expression is obtained by changing the sign of the imaginary part.

For example, the conjugate of(5x+4i) is (5x-4i), or
the conjugate of (2-3i) is (2+3i).

By multiplying the numerator and denominator by the complex conjugate, the imaginary part of the denominator will be eliminated.

To simplify (i-3)/(4-i), we multiply both the numerator and denominator by the conjugate of (4-i), i.e. (4+i) to give
(i-3)/(4-i)
=(i-3)(4+i)/((4-i)(4+i))
=(i²+i-12)/(4²-i²)
=(-1+i-12)/(16-(-1))
=(i-13)/17

For the given problem, you will need to specify more clearly the expression to avoid ambiguity. It could be interpreted as one of the two following cases:
A.
1 + i over i - 3 over 4 - i
= (1+i) / ((i-3)(4-i))
or
B.
1 + i over i - 3 over 4 - i
= (1+i) / ((i-3)/(4-i))

Once the expression is determined, you can proceed to simplify accordingly.