What is the multiplicative inverse of 6-3i?

The multiplicative inverse is the fraction whose numerator
is 1 and denominator is the given complex number.

We'll
note the inverse as x:

(6-3i)*x =
1

We'll divide by (6-3i) both
sides:

x = 1/(6-3i)

Since it
is not allowed to keep a complex number to denominator, we'll multiply the ratio by the
conjugate of the complex number:

x =
(6+3i)/(6-3i)(6+3i)

We'll re-write the denominator as a
difference of squares:

(6-3i)(6+3i) = 6^2 -
(3i)^2

(6-3i)(6+3i) = 36 -
9i^2

But i^2 =
-1:

(6-3i)(6+3i) = 36 +
9

(6-3i)(6+3i) =  45

We'll
re-write x:

x = (6+3i)/45

x =
6/45 + 3i/45

We'll simplify and we'll
get:

x =  2/15 +
i/15

The multiplicative inverse of the
complex number 6 - 3i is 2/15 + i/15.