Call the radicals $X$, $Y$ and $Z$ for simplicity.
Case 1: all the same (XXXXXX, YYYYYY or ZZZZZZ)
one way for each case:
Total for Case 1 = 3 ways
Case 2: just two radicals used, e.g., (X and Y) or (X and Z) or (Y and Z)
For each pair of radicals used:
5X 1Y: one way
4X 2Y: three ways
3X 3Y: four ways
2X 4Y: three ways
1X 5Y: one way
total for each pair: 12 ways.
Three choices for pairs so...
Total for Case 2 = 36 ways
Case 3: Three radicals: X and Y and Z
subcase a: 2 X, 2 Y, 2 Z
subcase b: 1 X
[I hope you get the principle and can complete this]
In the third case you must use all three radicals, X and Y and Z.
Either each is represented twice (XX and YY and ZZ)
...or not. One of the radicals must appear just once. Suppose it is X.
Then you have:
for each of these you can enumerate the distinct cases.