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Click to Select a Page... 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 Fun With Math Strip Home Page DavesFunStuff Dave's Fun Stuff Find your favorite TV shows with "Let's Watch TV!" Dave's Fun Stuff SDN Media News and More Dave's Fun Stuff Super Birthdays Contact Webmaster

Formula Summary So Far II

So far, we tested this Superpowerition operation using several formulas on positive and negative integers and decimal numbers including the number zero. That very much sums up our exploration of the Superpowerition operation in a nutshell. Let's summarize what we found so far.

Superpowerition operation notation expression: a ✱ b

a ✱ b Formulas

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))

Rules:

For the formula: a ✱ b

• This formula always works for all positive values of "a" and all values of "b" can be used as long as "a" is positive.
• 0 ✱ [any number] = undefined, so a≠0.
• "a" ✱ 1 = "a".
• 1 ✱ [any number] = 1.
• If "a" > 0, "a" ✱ 0 = ^(a)√(a) or a^{(1 ÷ a)}
• If "a" > 0 and an integer and "b" is an integer, the answer is always an integer, else, the answers may be decimal or irrational.
• If "a" is -1 and "b" is an integer, the answer is -1, else if "b" is decimal, then the answer is undefined.
• If "a" is a negative number and "b" is decimal, then the answer is always undefined.
• If "a" is a negative integer and "b" is a positive integer, then the answer is definable.
• If "a" is a negative integer less than -1 and "b" is 0 or less, then the answer is undefined.

Having a negative number in either "a" or "b" produces unpredictable results or an undefined answer. Therefore, it's best you stick to all positive values of "a" and any value of "b".

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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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