The solubility of Pb(OH)2 in water is 6.7 *10^-6 M.What is its solubility in a buffer solution of pH=8 ?

For the purposes of this question, the buffer solution may just imply that this pH is fairly constant. For a more precise answer if it doesn't, I would need the buffer constituents.

We're considering this equilibrium,

#Pb(OH)_2(s) rightleftharpoons Pb^(2+)(aq) + 2OH^(-)(aq)#

where #K_("sp") = [Pb^(2+)][OH^-]^2# e

#K_("sp") approx (6.7*10^-6)(2*6.7*10^-6)^2 approx 1.20*10^-15#

If #"pH" = 8#, poi #[OH^(-)] approx 10^-6#.

Focusing again on the equilibrium,

#Pb(OH)_2(s) rightleftharpoons Pb^(2+)(aq) + 2OH^(-)(aq)#

poi,

#1.20*10^-15 = x(10^-6+2x)^2#,
#therefore x approx 6.4*10^-6#

assuming that no interaction occurs with the buffer solution. Refer to the answer above if you need something more precise.