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Let's try out some variables for a using the formula a ^ (a ^ (b - 1)). We'll let b have the values from 1 to 4.
| −1 ✱ 1 = −1 ^ (−1 ^ (1 - 1)) = −1 ^ (−1 ^ ( 0)) = −1 ^ 1 = −1 |
| −1 ✱ 2 = −1 ^ (−1 ^ (2 - 1)) = −1 ^ (−1 ^ ( 1)) = −1 ^ −1 = −1 |
| −1 ✱ 3 = −1 ^ (−1 ^ (3 - 1)) = −1 ^ (−1 ^ ( 2)) = −1 ^ 1 = −1 |
| −1 ✱ 4 = −1 ^ (−1 ^ (4 - 1)) = −1 ^ (−1 ^ ( 3)) = −1 ^ −1 = −1 |
| −2 ✱ 1 = −2 ^ (−2 ^ (1 - 1)) = −2 ^ (−2 ^ ( 0)) = −2 ^ 1 = −2 |
| −2 ✱ 2 = −2 ^ (−2 ^ (2 - 1)) = −2 ^ (−2 ^ ( 1)) = −2 ^ −2 = 1/4 (0.25) |
| −2 ✱ 3 = −2 ^ (−2 ^ (3 - 1)) = −2 ^ (−2 ^ ( 2)) = −2 ^ 4 = 16 |
| −2 ✱ 4 = −2 ^ (−2 ^ (4 - 1)) = −2 ^ (−2 ^ ( 3)) = −2 ^ −8 = 1/256 (0.00390625) |
| −3 ✱ 1 = −3 ^ (−3 ^ (1 - 1)) = −3 ^ (−3 ^ ( 0)) = −3 ^ 1 = −3 |
| −3 ✱ 2 = −3 ^ (−3 ^ (2 - 1)) = −3 ^ (−3 ^ ( 1)) = −3 ^ −3 = −1/27 (−0.037037037037...) |
| −3 ✱ 3 = −3 ^ (−3 ^ (3 - 1)) = −3 ^ (−3 ^ ( 2)) = −3 ^ 9 = −19,683 |
| −3 ✱ 4 = −3 ^ (−3 ^ (4 - 1)) = −3 ^ (−3 ^ ( 3)) = −3 ^ −27 = −1.3113726 × 10−13 |
| −4 ✱ 1 = −4 ^ (−4 ^ (1 - 1)) = −4 ^ (−4 ^ ( 0)) = −4 ^ 1 = −4 |
| −4 ✱ 2 = −4 ^ (−4 ^ (2 - 1)) = −4 ^ (−4 ^ ( 1)) = −4 ^ −4 = 1/256 (0.00390625) |
| −4 ✱ 3 = −4 ^ (−4 ^ (3 - 1)) = −4 ^ (−4 ^ ( 2)) = −4 ^ 16 = 4,294,967,296 |
| −4 ✱ 4 = −4 ^ (−4 ^ (4 - 1)) = −4 ^ (−4 ^ ( 3)) = −4 ^ −64 = 2.9387358 × 10−39 |
You may have noticed that if a is -1, no matter what b is, the answer is −1.
If a is -2 or lower, the values of b for 1 through 4 jump from one extreme to another in range.
As b gets higher, the answers in the equation go from one far extreme to another, with numbers ranging from small to large to very small to very large to ultra small to ultra large and so forth. Compared with exponents, superpoweriton operations have a way of making the extremes of the answers wider sooner.
What if the value of b was zero or less? Turn to the next page.
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