how do you find the geometric mean of a pair of numbers and put it in the simplest radical form? (e.g. 3 and 9)

The geometric mean of of any set of N numbers is just the nth root of their product =

ⁿ√(a*b*c*d*...*n)

Do you mean like "3,5,6,7,9,1 = (Mean) 5.6666666661 = 6?

I think that it is         3,x,9

$$x/3=9/x

x^2=27

x=
\sqrt{27}=3\sqrt3$$

How do you find the geometric mean of a pair of numbers and put it in the simplest radical form? (e.g. 3 and 9)

For instance, the geometric mean of two numbers, say 2 and 8,

is just the square root of their product; that is $$\sqrt{2\cdot 8}=4$$.

$$\small{\text{
The geomeric mean of the pair of numbers 3 and 9 is:
$
\sqrt[2]{3*9}= \sqrt[2]{3}* \sqrt[2]{9}= \sqrt[2]{3}* 3
$
}}$$

See: https://en.wikipedia.org/wiki/Geometric_mean