Let's try out some variables for a using the formula a ^ (a ^ (b - 1)). We'll let b have the values from 2 to -4 and a be the value of -1.
−1 ✱ 2 = −1 ^ (−1 ^ (2 - 1)) = −1 ^ (−1 ^ ( 1)) = −1 ^ −1 = −1 |
−1 ✱ 1 = −1 ^ (−1 ^ (1 - 1)) = −1 ^ (−1 ^ ( 0)) = −1 ^ 1 = −1 |
−1 ✱ 0 = −1 ^ (−1 ^ (0 - 1)) = −1 ^ (−1 ^ (−1)) = −1 ^ −1 = −1 |
−1 ✱ −1 = −1 ^ (−1 ^ (−1 - 1)) = −1 ^ (−1 ^ (−2)) = −1 ^ 1 = −1 |
−1 ✱ −2 = −1 ^ (−1 ^ (−2 - 1)) = −1 ^ (−1 ^ (−3)) = −1 ^ −1 = −1 |
−1 ✱ −3 = −1 ^ (−1 ^ (−3 - 1)) = −1 ^ (−1 ^ (−4)) = −1 ^ 1 = −1 |
−1 ✱ −4 = −1 ^ (−1 ^ (−4 - 1)) = −1 ^ (−1 ^ (−5)) = −1 ^ −1 = −1 |
As you can see, raising negative 1 to an odd power is negative 1, and raising negative 1 to an even power including zero is positive 1. As both of those values are used as expnents to raise the base of negative 1 to a power, the result is always going to be negative 1.
Therefore, −1 ✱ [any number] = −1.
−2 ✱ 2 = −2 ^ (−2 ^ (2 - 1)) = −2 ^ (−2 ^ ( 1)) = −2 ^ −2 = 1/4 (0.25) |
−2 ✱ 1 = −2 ^ (−2 ^ (1 - 1)) = −2 ^ (−2 ^ ( 0)) = −2 ^ 1 = −2 |
−2 ✱ 0 = −2 ^ (−2 ^ (0 - 1)) = −2 ^ (−2 ^ (−1)) = −2 ^ (−1/2) = undefined |
−2 ✱ −1 = −2 ^ (−2 ^ (−1 - 1)) = −2 ^ (−2 ^ (−2)) = −2 ^ (−1/4) = undefined |
−2 ✱ −2 = −2 ^ (−2 ^ (−2 - 1)) = −2 ^ (−2 ^ (−3)) = −2 ^ (−1/8) = undefined |
−2 ✱ −3 = −2 ^ (−2 ^ (−3 - 1)) = −2 ^ (−2 ^ (−4)) = −2 ^ (−1/16) = undefined |
−2 ✱ −4 = −2 ^ (−2 ^ (−4 - 1)) = −2 ^ (−2 ^ (−5)) = −2 ^ (−1/32) = undefined |
−3 ✱ 2 = −3 ^ (−3 ^ (2 - 1)) = −3 ^ (−3 ^ ( 1)) = −3 ^ −3 = −1/27 (−0.037037037037...) |
−3 ✱ 1 = −3 ^ (−3 ^ (1 - 1)) = −3 ^ (−3 ^ ( 0)) = −3 ^ 1 = −3 |
−3 ✱ 0 = −3 ^ (−3 ^ (0 - 1)) = −3 ^ (−3 ^ (−1)) = −3 ^ (−1/3) = undefined |
−3 ✱ −1 = −3 ^ (−3 ^ (−1 - 1)) = −3 ^ (−3 ^ (−2)) = −3 ^ (−1/9) = undefined |
−3 ✱ −2 = −3 ^ (−3 ^ (−2 - 1)) = −3 ^ (−3 ^ (−3)) = −3 ^ (−1/27) = undefined |
−3 ✱ −3 = −3 ^ (−3 ^ (−3 - 1)) = −3 ^ (−3 ^ (−4)) = −3 ^ (−1/81) = undefined |
−3 ✱ −4 = −3 ^ (−3 ^ (−4 - 1)) = −3 ^ (−3 ^ (−5)) = −3 ^ (−1/243) = undefined |
−4 ✱ 2 = −4 ^ (−4 ^ (2 - 1)) = −4 ^ (−4 ^ ( 1)) = −4 ^ −4 = 1/256 (0.00390625) |
−4 ✱ 1 = −4 ^ (−4 ^ (1 - 1)) = −4 ^ (−4 ^ ( 0)) = −4 ^ 1 = −4 |
−4 ✱ 0 = −4 ^ (−4 ^ (0 - 1)) = −4 ^ (−4 ^ (−1)) = −4 ^ (−1/4) = undefined |
−4 ✱ −1 = −4 ^ (−4 ^ (−1 - 1)) = −4 ^ (−4 ^ (−2)) = −4 ^ (−1/16) = undefined |
−4 ✱ −2 = −4 ^ (−4 ^ (−2 - 1)) = −4 ^ (−4 ^ (−3)) = −4 ^ (−1/64) = undefined |
−4 ✱ −3 = −4 ^ (−4 ^ (−3 - 1)) = −4 ^ (−4 ^ (−4)) = −4 ^ (−1/256) = undefined |
−4 ✱ −4 = −4 ^ (−4 ^ (−4 - 1)) = −4 ^ (−4 ^ (−5)) = −4 ^ (−1/1024) = undefined |
Well, it looks like the negative numbers for variable b has failed, except when variable a is negative 1.
Please turn to the next page for the Formula Summary So Far.
© 1995-2023. davesfunstuff.com. All Rights Reserved. Reproduction of any part of this website without expressed written consent is prohibited.