Fun With Math-Superpowerition: Inventing a Super Power Operation -- a ✱ 0

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a ✱ 0

Now, let's look at the Superpowerition operation using the number "0". Let's begin by placing the "0" to the right of the superpowerition notation symbol:

a ✱ 0

Let's try out some variables for a using the formula a ^ (a ^ (b - 1)) as it becomes a ^ (a ^ (0 - 1)), then a ^ (a ^ −1)

1 ✱ 0 = 1 ^ ( 1 ^ (0 - 1)) = 1 ^ ( 1 ^ −1) = 1 ^ 1 = 1

2 ✱ 0 = 2 ^ ( 2 ^ (0 - 1)) = 2 ^ ( 2 ^ −1) = 2 ^ (1/2) ≈ 1.4142135 (the square root of 2)

3 ✱ 0 = 3 ^ ( 3 ^ (0 - 1)) = 2 ^ ( 3 ^ −1) = 3 ^ (1/3) ≈ 1.4422495 (the third root of 3)

4 ✱ 0 = 4 ^ ( 4 ^ (0 - 1)) = 4 ^ ( 4 ^ −1) = 4 ^ (1/4) ≈ 1.4142135 (the fourth root of 4)

5 ✱ 0 = 5 ^ ( 5 ^ (0 - 1)) = 5 ^ ( 5 ^ −1) = 5 ^ (1/5) ≈ 1.3797296 (the fifth root of 5)

9 ✱ 0 = 9 ^ ( 9 ^ (1 - 1)) = 9 ^ ( 9 ^ −1) = 9 ^ (1/9) ≈ 1.2765180 (the 9th root of 9)

16 ✱ 0 = 16 ^ (16 ^ (1 - 1)) = 16 ^ (16 ^ −1) = 16 ^ (1/16) ≈ 1.1892071 (the 16th root of 16)

As you can see, a ✱ 0 yields the a root of a (where a > 0) which is represented by ^a√(a).

Therefore, a ✱ 0 = ^(a)√(a) or a^{(1 ÷ a)}.

A few tips. When you do a square root of a number, you're raising the number to the power of the reciprocal of the square root, so taking the 2nd root of a number is the same as raising the number exponentially by 1/2.

Here's a partial chart:

x ^ (1/2) = x ROOT 2 (2nd root of x)
x ^ (1/3) = x ROOT 3 (3rd root of x)
x ^ (1/4) = x ROOT 4 (4th root of x)
x ^ (1/5) = x ROOT 5 (5th root of x)

The expression a ✱ b, which is translated to a ^ (a ^ (b - 1)), can also be translated to this:

(a ^ (a ^ b)) ROOT a or ^a√(a ^ (a ^ b))

Using both formulas, we'll perform six Superpowerition operations using six pairs of two numbers:

2 ✱ 0 =
2 ^ (2 ^ (0 - 1)) = 2 ^ (2 ^ -1) = 2 ^ (1/2) ≈ 1.4142135
(2 ^ (2 ^ 0)) ROOT 2 = (2 ^ 1) ROOT 2 = 2 ROOT 2 ≈ 1.4142135

2 ✱ 3 =
2 ^ (2 ^ (3 - 1)) = 2 ^ (2 ^ 2) = 2 ^ 4 = 16
(2 ^ (2 ^ 3)) ROOT 2 = (2 ^ 8) ROOT 2 = (256) ROOT 2 = 16

3 ✱ 0 =
3 ^ (3 ^ (0 - 1)) = 3 ^ (3 ^ -1) = 3 ^ (1/3) ≈ 1,4422495
(3 ^ (3 ^ 0)) ROOT 3 = (3 ^ 1) ROOT 3 = 3 ROOT 3 ≈ 1.4422495

3 ✱ 4 =
3 ^ (3 ^ (4 - 1)) = 3 ^ (3 ^ 3) = 3 ^ 27 = 7,625,597,484,987
(3 ^ (3 ^ 4)) ROOT 3 = (3 ^ 81) ROOT 3 = (4.4342648 X 10 ^ 38) ROOT 3 = 7,625,597,484,987

4 ✱ 0 =
4 ^ (4 ^ (0 - 1)) = 4 ^ (4 ^ -1) = 4 ^ (1/4) ≈ 1.4142135
(4 ^ (4 ^ 0)) ROOT 4 = (4 ^ 1) ROOT 4 = 4 ROOT 4 ≈ 1.4142135

4 ✱ 3 =
4 ^ (4 ^ (3 - 1)) = 4 ^ (4 ^ 2) = 4 ^ 16 = 4,294,967,296
(4 ^ (4 ^ 3)) ROOT 4 = (4 ^ 64) ROOT 4 = (4 ^ 64) ROOT 4 = (3.4028236 X 10 ^ 38) ROOT 4 = 4,294,967,296

a ✱ 0

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