The fifth term of a geometric sequence of positive numbers is $11$ and the eleventh term is $5$. What is the eighth term of the sequence? Express your answer in simplest radical form.

Let a1 be the first term

So....we have that

11 / 5 = a1 (r)^4 / [ a1 (r)^10]

11/5 = r^4/r^10

11/5 = 1/ r^6

r^6 = (5/11)

r = (5/11)^(1/6)

So...the eighth term is

11 * [(5/11)^(1/6)]^3 =

11 ( 5/11)^(1/2) =

11 √5/ √11 =

√11*√5 =

√55