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This is based on the What is a Word/Phrase™ series of Phrase™ and Word™ puzzles, started by JLee.

If a word follows a certain rule, then I call it a Powerful Word™. If it violates this rule, then it is not a Powerful Word™.

Use the example word lists below to find the rule.

Powerful Word™, Not Powerful Word™
DIRECTORSHIP, INTELLECT
MONUMENTALISM, WIDTH
ACE, KING
FADE, DISAPPEAR
NONPRODUCTIVITY, ROT
GAFFE, ABERRATION
BEHELD, OBSERVED
ANNIHILATOR, BUILDER
ICEBERG, WHIRLPOOL
RELIABLE, UNSTOPPABLE
ZERO, INFINITY
PREDICATE, SUBJECT
RESEARCHER, SCIENTIST
COUNTERCULTURE, FOLK
OAK, REDWOOD

Each word can be tested for whether it is a Powerful Word™ without depending on other words.

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Mekalikot has paved the way. A Powerful Word is ...

... a word, whose letter sum is its length raised to the power of a natural number. Usually, it's the square, but ZERO and OAK are cubes. There are words with square letters ums in the list of Non-powerful words, but these are not squares of their word length.

The letter sum is the sum of all letters, converted to their position in the alphabet: A→1, B→2, C→3, ..., Z→26. For example: ZERO → 26 + 5 + 18 + 15 = 64 = 4³.

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  • $\begingroup$ Clever how the word length is also used. $\endgroup$ – Takeshi Jan 13 '17 at 7:13
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[EDIT]Partially wrong

Powerful Word™ is:

a word that if you convert every letter to a number and get its sum, will result to the product of a Perfect Square(the product of a rational number multiplied by itself) or "a number, raised to the second power"

Guide:

A=1, B=2 and so on...

1. DIRECTORSHIP = 144 = $12^2 $
2. MONUMENTALISM = 169 = $13^2 $
3. ACE = 9 = $3^2 $
4. FADE = 16 = $4^2 $
5. NONPRODUCTIVITY = 225 = $15^2 $
6. GAFFE = 25 = $5^2 $
7. BEHELD = 36 = $6^2 $
8. ANNIHILATOR = 121 = $11^2 $
9. ICEBERG = 49 = $7^2 $
10. RELIABLE = 64 = $8^2 $
11. ZERO = 64 = $8^2 $
12. PREDICATE = 81 = $9^2 $
13. RESEARCHER = 100 = $10^2 $
14. COUNTERCULTURE = 196 = $14^2 $
15. OAK = 27 = power of 3

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    $\begingroup$ I'm confused. Why is OAK 84? Shouldn't it be only 27? 15 + 1 + 11? 27 which is a 3 raised to 3. This will still be a POWERFUL word but it doesn't only apply to perfect squares. $\endgroup$ – Takeshi Jan 13 '17 at 3:20
  • $\begingroup$ This answer is on the right track, but incorrect. Did you check both lists? $\endgroup$ – MikeQ Jan 13 '17 at 3:38
  • $\begingroup$ oops, yeah, will edit OAK. @MikeQ, ok will check again $\endgroup$ – Mekalikot Jan 13 '17 at 3:52

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