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

Decimal -2 < a < 0 ✱ Decimal -1 < b < 2

On this page, we'll let "a" have the value of -2, -1.5, -1 and -0.5 (0 produces and undefined result). We'll also let "b" have the values of -1, 0.5, 0, 0.5, 1, 1.5 and 2.

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

-2.00 ✱ -1.00 = -2.00 ^ (-2.00 ^ (-1.00 - 1)) = -2.00 ^ (-2.00 ^ (-2.00)) = -2.00 ^ 0.25 = undefined
-1.50 ✱ -1.00 = -1.50 ^ (-1.50 ^ (-1.00 - 1)) = -1.50 ^ (-1.50 ^ (-2.00)) ≈ -1.50 ^ 0.4444444 = undefined
-1.00 ✱ -1.00 = -1.00 ^ (-1.00 ^ (-1.00 - 1)) = -1.00 ^ (-1.00 ^ (-2.00)) = -1.00 ^ 1 = -1
-0.50 ✱ -1.00 = -0.50 ^ (-0.50 ^ (-1.00 - 1)) = -0.50 ^ (-0.50 ^ (-2.00)) = -0.50 ^ 4 = 0.625
-2.00 ✱ -0.50 = -2.00 ^ (-2.00 ^ (-0.50 - 1)) = -2.00 ^ (-2.00 ^ (-1.50)) = -2.00 ^ undefined = undefined
-1.50 ✱ -0.50 = -1.50 ^ (-1.50 ^ (-0.50 - 1)) = -1.50 ^ (-1.50 ^ (-1.50)) = -1.50 ^ undefined = undefined
-1.00 ✱ -0.50 = -1.00 ^ (-1.00 ^ (-0.50 - 1)) = -1.00 ^ (-1.00 ^ (-1.50)) = -1.00 ^ undefined = undefined
-0.50 ✱ -0.50 = -0.50 ^ (-0.50 ^ (-0.50 - 1)) = -0.50 ^ (-0.50 ^ (-1.50)) = -0.50 ^ undefined = undefined
-2.00 ✱ 0.00 = -2.00 ^ (-2.00 ^ ( 0.00 - 1)) = -2.00 ^ (-2.00 ^ (-1.00)) = -2.00 ^ -0.5 = undefined
-1.50 ✱ 0.00 = -1.50 ^ (-1.50 ^ ( 0.00 - 1)) = -1.50 ^ (-1.50 ^ (-1.00)) = -1.50 ^ -0.6666666 = undefined
-1.00 ✱ 0.00 = -1.00 ^ (-1.00 ^ ( 0.00 - 1)) = -1.00 ^ (-1.00 ^ (-1.00)) = -1.00 ^ -1 = -1
-0.50 ✱ 0.00 = -0.50 ^ (-0.50 ^ ( 0.00 - 1)) = -0.50 ^ (-0.50 ^ (-1.00)) = -0.50 ^ -2 = 4
-2.00 ✱ 0.50 = -2.00 ^ (-2.00 ^ ( 0.50 - 1)) = -2.00 ^ (-2.00 ^ (-0.50)) = -2.00 ^ undefined = undefined
-1.50 ✱ 0.50 = -1.50 ^ (-1.50 ^ ( 0.50 - 1)) = -1.50 ^ (-1.50 ^ (-0.50)) = -1.50 ^ undefined = undefined
-1.00 ✱ 0.50 = -1.00 ^ (-1.00 ^ ( 0.50 - 1)) = -1.00 ^ (-1.00 ^ (-0.50)) = -1.00 ^ undefined = undefined
-0.50 ✱ 0.50 = -0.50 ^ (-0.50 ^ ( 0.50 - 1)) = -0.50 ^ (-0.50 ^ (-0.50)) = -0.50 ^ undefined = undefined
-2.00 ✱ 1.00 = -2.00 ^ (-2.00 ^ ( 1.00 - 1)) = -2.00 ^ (-2.00 ^ ( 0.00)) = -2.00 ^ 1 = -2
-1.50 ✱ 1.00 = -1.50 ^ (-1.50 ^ ( 1.00 - 1)) = -1.50 ^ (-1.50 ^ ( 0.00)) = -1.50 ^ 1 = -1.5
-1.00 ✱ 1.00 = -1.00 ^ (-1.00 ^ ( 1.00 - 1)) = -1.00 ^ (-1.00 ^ ( 0.00)) = -1.00 ^ 1 = -1
-0.50 ✱ 1.00 = -0.50 ^ (-0.50 ^ ( 1.00 - 1)) = -0.50 ^ (-0.50 ^ ( 0.00)) = -0.50 ^ 1 = -0.5
-2.00 ✱ 1.50 = -2.00 ^ (-2.00 ^ ( 1.50 - 1)) = -2.00 ^ (-2.00 ^ ( 0.50)) = -2.00 ^ undefined = undefined
-1.50 ✱ 1.50 = -1.50 ^ (-1.50 ^ ( 1.50 - 1)) = -1.50 ^ (-1.50 ^ ( 0.50)) = -1.50 ^ undefined = undefined
-1.00 ✱ 1.50 = -1.00 ^ (-1.00 ^ ( 1.50 - 1)) = -1.00 ^ (-1.00 ^ ( 0.50)) = -1.00 ^ undefined = undefined
-0.50 ✱ 1.50 = -0.50 ^ (-0.50 ^ ( 1.50 - 1)) = -0.50 ^ (-0.50 ^ ( 0.50)) = -0.50 ^ undefined = undefined
-2.00 ✱ 2.00 = -2.00 ^ (-2.00 ^ ( 2.00 - 1)) = -2.00 ^ (-2.00 ^ ( 1.00)) = -2.00 ^ -2 = 0.25
-1.50 ✱ 2.00 = -1.50 ^ (-1.50 ^ ( 2.00 - 1)) = -1.50 ^ (-1.50 ^ ( 1.00)) = -1.50 ^ -1.5 = undefined
-1.00 ✱ 2.00 = -1.00 ^ (-1.00 ^ ( 2.00 - 1)) = -1.00 ^ (-1.00 ^ ( 1.00)) = -1.00 ^ -1 = -1
-0.50 ✱ 2.00 = -0.50 ^ (-0.50 ^ ( 2.00 - 1)) = -0.50 ^ (-0.50 ^ ( 1.00)) = -0.50 ^ -0.5 = undefined

As you can see, if the "a" value in "a ✱ b" is negative, it produces unpredictable results. If the "b" value is a decimal when the "a" value is negative, the answers are always going to be undefined. If the "b" value is 1, then the answer is going to be the value of "a". All other values of "a" and "b" depend on how their values work out in the formula above. You can't take a negative number (decimal or integer) to a negative decimal number as you can see in the table above.

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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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