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You are at the section Fun With Math-Superpowerition: Inventing a Super Power Operation

Formula Summary So Far

So far, we tested this Superpowerition operation using several formulas on positive integers, negative integers, and zero in both variables a and b in the operation. Let's summarize what we found so far.

Superpowerition operation notation expression: a ✱ b

a ✱ b Formulas

FormulaCodingSame asSame 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))

Special Notes: (applies when a and b are integer values only for now)

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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Fun With Math-Superpowerition: Inventing a Super Power Operation Main Page Introduction Comparing The Math Operators I Comparing The Math Operators II Using Numbers 3 and Higher 1 ✱ b a ✱ 1 a ✱ 0 0 ✱ b a ✱ -b -a ✱ b -a ✱ -b Formula Summary So Far Integer a > 1 ✱ Decimal b > 1 Integer a > 1 ✱ Decimal 0 < b < 1 Integer a > 1 ✱ Decimal b < 0 Decimal a > 1 ✱ Decimal b > 1 Decimal a > 1 ✱ Decimal 0 < b < 1 Decimal a > 1 ✱ Decimal b < 0 Decimal 0 < a < 1 ✱ Decimal b > 1 Decimal 0 < a < 1 ✱ Decimal 0 < b < 1 Decimal 0 < a < 1 ✱ Decimal b < 0 Decimal -2 < a < 0 ✱ Decimal -1 < b < 2 Formula Summary So Far II Why 2 ✱ 0 is Not 1 Finding The Inverse Operations 1.25 ✱ -10 to 10 1.5 ✱ -10 to 10 1.75 ✱ -10 to 10 2 ✱ -10 to 10 2.5 ✱ -10 to 10 3 ✱ -10 to 10 3.5 ✱ -10 to 10 4 ✱ -10 to 10 4.5 ✱ -10 to 10 5 ✱ -10 to 10 6 ✱ -10 to 10 7 ✱ -10 to 10 8 ✱ -10 to 10 9 ✱ -10 to 10 10 ✱ -10 to 10
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