Given these numbers, find the hidden word

Question

Given this sorted list of 33 numbers, is it in your power to find the riddle they are hiding and figure out the secret word?

1
3
4
6
9
9
9
18
23
25
25
64
125
169
196
225
225
625
625
5,832
8,000
10,648
16,384
17,576
38,416
78,125
279,841
390,625
1,889,568
1,889,568
34,012,224
47,045,881
16,983,563,041

Hint:

_ _ _ _ _ _ _ _ _ _ _ _ _ ’ _ _ _ _ _ ,
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

---

Additional hints added after @GarethMcCaughan solved the puzzle:

Hint 2:

A helpful tool: https://www.wolframalpha.com/

Hint 3:

$V^3 = 10,648$

Answer 1

The numbers

are mostly powers of smaller numbers -- and in particular each of them is a power of a number from 1 to 26.

If we

choose wisely in ambiguous cases (e.g., 9 = 3^2 = 9^1)

then

we can take a^b to mean "there is an instance of letter a in position b in some word", where the letters are converted in the usual A1Z26 way

and

assign letters to words so as to get the message (using the letter-counts in the hint -- I think this would probably be too difficult without that) WHEN FORWARDSS I'M COZY, / IN REVERSE I DETER.

And I think the secret word is

SNUG, meaning cozy; reversed we get GUNS which may indeed be used to deter.

Confession:

I initially had REVERSES and FORWARD, slightly garbling the riddle but leaving it still solvable (with SNUG and GUNS the other way around); OP kindly pointed out my mistake.