The multiplicative inverse of a number A is another number B which when multiplied by A gives 1.

Here we have the number 3 + 2i

Let A = 3 + 2i, we need to find B such that A*B = 1

=> B = 1/(3 + 2i)

=> B = (3 - 2i)/(3 + 2i)(3 - 2i)

=> 3 - 2i /( 9 + 4)

=> 3/13 - 2i/13

**The multiplicative inverse is 3/13 - 2i/13**

We'll get the multiplicative inverse when multiplying the given complex number by the inverse number we'll get the result 1.

We'll note the inverse as x:

(3 + 2i)*x = 1

We'll divide by (3 + 2i) both sides:

x = 1/(3 + 2i)

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

x = (3 - 2i)/(3 + 2i)(3 - 2i)

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

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

(3 + 2i)(3 - 2i) = 9 - 4i^2

But i^2 = -1:

(3 + 2i)(3 - 2i) = 9 + 4

(3 + 2i)(3 - 2i) = 13

We'll re-write x:

x = (3 - 2i)/13

x = 3/13 - 2i/13

**The multiplicative inverse of the complex number 3 + 2i is 3/13 - 2i/13.**