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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