Superpowerition operation notation expression: a ✱ b
Formula | Coding | Same as | Same as |
---|---|---|---|
1 | a ^ (1 / (a ^ ((b × −1) + 1))) | a ↑ (1 ÷ (a ↑ ((b × −1) + 1))) | a(1 ÷ (a((b × −1) + 1))) |
2 | a ^ (a ^ (b - 1)) | a ↑ (a ↑ (b - 1)) | a(a(b - 1)) |
3 | N.A. | (a ^ (a ^ b)) ROOT a or (a ↑ (a ↑ b)) ↓ a |
a√(a ↑ (a ↑ b)) ora√(a(ab)) |
Also note that a√(b) or b ↓ a is the same as b(1/a) or b(1 ÷ a), so the above formula could be converted to this below: | 4 | (a ^ (a ^ b)) ^ (1 / a) | (a ↑ (a ↑ b)) ↑ (1 ÷ a) | (a(ab))(1 ÷ a) |
Because (ab)c = a(b × c), the above formula can be converted to this below: | 5 | a ^ ((a ^ b) * (1 / a)) | a ↑ ((a ↑ b) × (1 ÷ a)) | a((ab) × (1 ÷ a)) |
a ✱ 1 = a.
1 ✱ [any number] = 1.
a ✱ 0 = (a)√(a) or a(1 ÷ a).
−1 ✱ [any number] = −1.
0 ✱ [any number] = undefined, so a≠0.
a ✱ b [if b ≤ 0] results in irrational numbers as answers.
a ✱ b [if b ≥ 1] results in rational numbers as answers.
If a ≤ −2, then it is a must that b ≥ 1.
If a ≤ −2, and b ≤ 0, then the answers are undefined.
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