In[]:=
(Sqrt[r0^2]DiracDelta[-Sqrt[r0^2-rho^2]+z])/Sqrt[​​r0^2-rho^2]-(Sqrt[r0^2]DiracDelta[Sqrt[r0^2-rho^2]+z])/Sqrt[​​r0^2-rho^2]//FullSimplify
Out[]=
2
r0
DiracDelta-
2
r0
-
2
rho
+z
2
r0
-
2
rho
-
2
r0
DiracDelta
2
r0
-
2
rho
+z
2
r0
-
2
rho
In[]:=
FactorTogether
2
r0
DiracDelta-
2
r0
-
2
rho
+z
2
r0
-
2
rho
-
2
r0
DiracDelta
2
r0
-
2
rho
+z
2
r0
-
2
rho

Out[]=
1
2
r0
-
2
rho
2
r0
DiracDelta-
2
r0
-
2
rho
+z-DiracDelta
2
r0
-
2
rho
+z