Let z = √5+3i
Then

−

z

= √5−3i ... (i)
and |z|2 = (√5)2+(3)2 = 14 ... (ii)
∴ The multiplicative inverse of √5+3i is given by :
z−1 =

− z

|z|2

=

√5−3i

14

(By using (i) and (ii))
=

√5

14

+i(

−3

14

)