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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

Decimal 0 < a < 1 ✱ Decimal b < 0

Now let's let a be a decimal number between 0 and 1 and b be a decimal value between -1 and 0 using steps of 0.25.

Let's try out some variables for a using the formula a ^ (a ^ (b - 1)).

0.00 ✱ -1.00 = 0.00 ^ (0.00 ^ (-1.00 - 1)) = 0.00 ^ (0.00 ^ (-2.00)) = 0.00 ^ 0.00 = undefined

0.25 ✱ -1.00 = 0.25 ^ (0.25 ^ (-1.00 - 1)) = 0.25 ^ (0.25 ^ (-2.00)) = 0.25 ^ 16 ≈ 2.3283064 × 10-10

0.50 ✱ -1.00 = 0.50 ^ (0.50 ^ (-1.00 - 1)) = 0.50 ^ (0.50 ^ (-2.00)) ≈ 0.50 ^ 4 = 0.0625

0.75 ✱ -1.00 = 0.75 ^ (0.75 ^ (-1.00 - 1)) = 0.75 ^ (0.75 ^ (-2.00)) ≈ 0.75 ^ 1.7777777 ≈ 0.5996346

1.00 ✱ -1.00 = 1.00 ^ (1.00 ^ (-1.00 - 1)) = 1.00 ^ (1.00 ^ (-2.00)) = 1.00 ^ 1.00 = 1.00

0.00 ✱ -0.75 = 0.00 ^ (0.00 ^ (-0.75 - 1)) = 0.00 ^ (0.00 ^ (-1.75)) = 0.00 ^ 0.00 = undefined

0.25 ✱ -0.75 = 0.25 ^ (0.25 ^ (-0.75 - 1)) = 0.25 ^ (0.25 ^ (-1.75)) ≈ 0.25 ^ 11.3137085 ≈ 0.000000154

0.50 ✱ -0.75 = 0.50 ^ (0.50 ^ (-0.75 - 1)) = 0.50 ^ (0.50 ^ (-1.75)) ≈ 0.50 ^ 3.3635856 ≈ 0.0971538

0.75 ✱ -0.75 = 0.75 ^ (0.75 ^ (-0.75 - 1)) = 0.75 ^ (0.75 ^ (-1.75)) ≈ 0.75 ^ 1.6544086 ≈ 0.6212984

1.00 ✱ -0.75 = 1.00 ^ (1.00 ^ (-0.75 - 1)) = 1.00 ^ (1.00 ^ (-1.75)) = 1.00 ^ 1.00 = 1.00

0.00 ✱ -0.50 = 0.00 ^ (0.00 ^ (-0.50 - 1)) = 0.00 ^ (0.00 ^ (-1.50)) = 0.00 ^ 0.00 = undefined

0.25 ✱ -0.50 = 0.25 ^ (0.25 ^ (-0.50 - 1)) = 0.25 ^ (0.25 ^ (-1.50)) ≈ 0.25 ^ 8 ≈ 0.000015258

0.50 ✱ -0.50 = 0.50 ^ (0.50 ^ (-0.50 - 1)) = 0.50 ^ (0.50 ^ (-1.50)) ≈ 0.50 ^ 2.8284271 ≈ 0.1407857

0.75 ✱ -0.50 = 0.75 ^ (0.75 ^ (-0.50 - 1)) = 0.75 ^ (0.75 ^ (-1.50)) ≈ 0.75 ^ 1.5396007 ≈ 0.6421614

1.00 ✱ -0.50 = 1.00 ^ (1.00 ^ (-0.50 - 1)) = 1.00 ^ (1.00 ^ (-1.50)) = 1.00 ^ 1.00 = 1.00

0.00 ✱ -0.25 = 0.00 ^ (0.00 ^ (-0.25 - 1)) = 0.00 ^ (0.00 ^ (-1.25)) = 0.00 ^ 0.00 = undefined

0.25 ✱ -0.25 = 0.25 ^ (0.25 ^ (-0.25 - 1)) = 0.25 ^ (0.25 ^ (-1.25)) ≈ 0.25 ^ 5.6568542 ≈ 0.000392856

0.50 ✱ -0.25 = 0.50 ^ (0.50 ^ (-0.25 - 1)) = 0.50 ^ (0.50 ^ (-1.25)) ≈ 0.50 ^ 2.3784142 ≈ 0.1923206

0.75 ✱ -0.25 = 0.75 ^ (0.75 ^ (-0.25 - 1)) = 0.75 ^ (0.75 ^ (-1.25)) ≈ 0.75 ^ 1.4327599 ≈ 0.6622055

1.00 ✱ -0.25 = 1.00 ^ (1.00 ^ (-0.25 - 1)) = 1.00 ^ (1.00 ^ (-1.25)) = 1.00 ^ 1.00 = 1.00

0.00 ✱ 0.00 = 0.00 ^ (0.00 ^ ( 0.00 - 1)) = 0.00 ^ (0.00 ^ (-1.00)) = 0.00 ^ 0.00 = undefined

0.25 ✱ 0.00 = 0.25 ^ (0.25 ^ ( 0.00 - 1)) = 0.25 ^ (0.25 ^ (-1.00)) = 0.25 ^ 4 ≈ 0.0039062

0.50 ✱ 0.00 = 0.50 ^ (0.50 ^ ( 0.00 - 1)) = 0.50 ^ (0.50 ^ (-1.00)) = 0.50 ^ 2 = 0.25

0.75 ✱ 0.00 = 0.75 ^ (0.75 ^ ( 0.00 - 1)) = 0.75 ^ (0.75 ^ (-1.00)) ≈ 0.75 ^ 1.3333333 ≈ 0.6814202

1.00 ✱ 0.00 = 1.00 ^ (1.00 ^ ( 0.00 - 1)) = 1.00 ^ (1.00 ^ (-1.00)) = 1.00 ^ 1.00 = 1.00

Once again, 0.00 ✱ [any number] = undefined. Also, as the a in the expression "a ✱ b" gets smaller, the answer gets exponentially smaller.

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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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