sqrt(1/5) = sqrt(1)/sqrt(5) = 1/sqrt(5)
However, we usually don't want radicals on the denominator, so we need to "rationalize" the denominator by multiplying both the numerator and denominator by sqrt(5):
1/sqrt(5) * sqrt(5)/sqrt(5)
= sqrt(5)/[sqrt(5)*sqrt(5)]
= sqrt(5)/[sqrt(25)]
= sqrt(5)/5
the products of square roots are the square roots of the products, we can see this more explicitly if we represent squareroots as powers. So,
sqrt(3/2) * sqrt(5/6) = sqrt[(3/2)*(5/6)]
.............................= sqrt(15/12)
.............................= sqrt(15)/sqrt(12)
.............................= sqrt(5*3)/sqrt(3*4)
.............................= [sqrt(3)*sqrt(5)]/[sqrt(3)*sqrt(4)]
.............................= sqrt(5)/2
EDIT: Just in case you were wondering, sqrt(180)/12 was correct, however, it was not simplified
sqrt(180)/12 = sqrt(36*5)/12
..................= sqrt(36)*sqrt(5)/12
..................= 6sqrt(5)/12
..................= sqrt(5)/2