Calculate the multiplicative inverse of (3+4i)/(4-5i).

The multiplicative inverse  of the given ratio
is:

(4-5i)/(3+4i)

Now, because
it is not allowed to have a complex number at denominator, we'll multiply the ratio by
the conjugate of (3+4i).

(4-5i)/(3+4i) =
(4-5i)*(3-4i)/(3+4i)*(3-4i)

We'll remove the
brackets:

(4-5i)*(3-4i) = 12 - 16i - 15i - 20 = -8-31i =
-(8+31i)

(3+4i)*(3-4i) = (3)^2 - (4i)^2 = 9 + 16 =
25

(4-5i)/(3+4i) =
-(8+31i)/25

The multiplicative
inverse is:

-8/25 -
(31/25)*i