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# Exponentiation

(Redirected from B↑n)

Exponentiation with an integer exponent $\scriptstyle d \,$ is repetitive multiplication (a 3rd iteration "hyper-addition"): a given number $\scriptstyle b \,$ (called the base) is repeatedly multiplied by itself a number of times $\scriptstyle d \,$ (called the exponent); this is usually notated $\scriptstyle b^d \,$ and read "$\scriptstyle b \,$ exponent $\scriptstyle d \,$." For example, $\scriptstyle 7^4 \,=\, 7 \times 7 \times 7 \times 7 \,=\, 2401 \,$.

## Exponentiation

### Exponentiation operator

In most computer programming languages, and in TeX source, the caret character ^ is used as the exponentiation operator (e.g. b^d,) although sometimes two asterisk characters ** are used as the exponentiation operator (e.g. b**d,) implying a 2nd iteration "hyper-multiplication."

You may also use Knuth's up-arrow notation $\scriptstyle b{\uparrow}d \,$ to represent exponentiation.

### Exponentiation table

The columns of the table, with fixed exponent $\scriptstyle d \,$, are powers $\scriptstyle n^d \,$. The rows of the table, with fixed base $\scriptstyle b \,$, are exponentials $\scriptstyle b^n \,$. The diagonal of the table (entries in bold) are $\scriptstyle n^n \,$

Exponentiation table $\scriptstyle b^d\, (b \,\ge\, 0,\, d \,\ge\, 0) \,$
$b \,$ \ $d \,$ 0 1 2 3 4 5 6 7 8 9 10 11 12
0 1 [1] 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 1 2 4 8 16 32 64 128 256 512 1024 2048 4096
3 1 3 9 27 81 243 729 2187 6561 19683 59049 177147 531441
4 1 4 16 64 256 1024 4096 16384 65536 262144 1048576 4194304 16777216
5 1 5 25 125 625 3125 15625 78125 390625 1953125 9765625 48828125 244140625
6 1 6 36 216 1296 7776 46656 279936 1679616 10077696 60466176 362797056 2176782336
7 1 7 49 343 2401 16807 117649 823543 5764801 40353607 282475249 1977326743 13841287201
8 1 8 64 512 4096 32768 262144 2097152 16777216 134217728 1073741824 8589934592 68719476736
9 1 9 81 729 6561 59049 531441 4782969 43046721 387420489 3486784401 31381059609 282429536481
10 1 10 100 1000 10000 100000 1000000 10000000 100000000 1000000000 10000000000 100000000000 1000000000000
11 1 11 121 1331 14641 161051 1771561 19487171 214358881 2357947691 25937424601 285311670611 3138428376721
12 1 12 144 1728 20736 248832 2985984 35831808 429981696 5159780352 61917364224 743008370688 8916100448256

### Base and exponent

#### Exponent

When $\scriptstyle b \,$ is positive and $\scriptstyle d \,$ is negative, the exponentiations are the reciprocals of the exponentiations of $\scriptstyle b \,$ with exponent $\scriptstyle |d| \,$.

For example, the exponentials (base 2) $\scriptstyle \{2^{-i}\}_{i=0}^{\infty} \,$ give $\scriptstyle \big\{1, \tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{8}, \cdots\big\} \,$.

We have the following rules

• $\scriptstyle b^d \,$ with $\scriptstyle d \,<\, 0 \,$ is $\scriptstyle \frac{1}{b^{|d|}},\ b \,\neq\, 0 \,$.
• $\scriptstyle b^0 \,=\, 1 \,$ for any real, imaginary or complex $\scriptstyle b \,$ (including $\scriptstyle b \,=\, 0 \,$ if $\scriptstyle b^0 \,$ is interpreted as the empty product, e.g. 1.)
• $\scriptstyle b^1 \,=\, b \,$

#### 0^0

If $\scriptstyle n^0 \,$ is interpreted as the empty product, which equals the multiplicative identity, i.e. 1 for numbers, this should be the result for any $\scriptstyle n \,$, including 0.

In algebra, for the binomial expansion

$(1+x)^n = \sum_{i=0}^{n} \binom{n}{i} x^i \,$

we need the conventions

$0! = 1,\ 0^0 = 1 \,$

for the constant term $\scriptstyle \binom{n}{0} x^0 \,=\, \frac{n!}{n!0!} x^0 \,$ to be 1 for any value of $\scriptstyle x \,$, including $\scriptstyle x \,=\, 0 \,$.

In regards to $\scriptstyle 0^0 \,$, see 0^0 or the special case of zero to the zeroeth power.

### Powers

When the exponent $\scriptstyle d \,$ is fixed, the exponentiation operations are considered powers (n^d or n**d)

$n^d \,$

#### Table of powers

A sequence of integers $\scriptstyle \{ n^d \}_{n=0}^{\infty} \,$ is called "the powers to the degree $\scriptstyle d \,$." Some sequences of powers in the OEIS are given in the following table

Table of powers
$\scriptstyle d \,$ $\scriptstyle n^d,\, n \,\ge\, 0, \,$ sequences A-number
0[1] {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...} A000012$\scriptstyle (n) \,$
1 {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, ...} A000027$\scriptstyle (n) \,$
2 {0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, ...} A000290$\scriptstyle (n) \,$
3 {0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, ...} A000578$\scriptstyle (n) \,$
4 {0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, ...} A000583$\scriptstyle (n) \,$
5 {0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, 161051, 248832, 371293, 537824, 759375, 1048576, 1419857, 1889568, 2476099, ...} A000584$\scriptstyle (n) \,$
6 {0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 16777216, 24137569, ...} A001014$\scriptstyle (n) \,$
7 {0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 4782969, 10000000, 19487171, 35831808, 62748517, 105413504, 170859375, 268435456, ...} A001015$\scriptstyle (n) \,$
8 {0, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721, 100000000, 214358881, 429981696, 815730721, 1475789056, 2562890625, ...} A001016$\scriptstyle (n) \,$
9 {0, 1, 512, 19683, 262144, 1953125, 10077696, 40353607, 134217728, 387420489, 1000000000, 2357947691, 5159780352, 10604499373, 20661046784, ...} A001017$\scriptstyle (n) \,$
10 {0, 1, 1024, 59049, 1048576, 9765625, 60466176, 282475249, 1073741824, 3486784401, 10000000000, 25937424601, 61917364224, 137858491849, ...} A008454$\scriptstyle (n) \,$
11 {0, 1, 2048, 177147, 4194304, 48828125, 362797056, 1977326743, 8589934592, 31381059609, 100000000000, 285311670611, 743008370688, ...} A008455$\scriptstyle (n) \,$
12 {0, 1, 4096, 531441, 16777216, 244140625, 2176782336, 13841287201, 68719476736, 282429536481, 1000000000000, 3138428376721, 8916100448256, ...} A008456$\scriptstyle (n) \,$

#### Powers as figurate numbers

Powers $\scriptstyle n^d \,$ may be considered as $\scriptstyle d \,$-dimensional regular orthotopic numbers.

Regular orthotopic numbers
$d \,$ $\scriptstyle d \,$-dimensional orthotopic numbers
0 Point numbers
1 Segment numbers (Cf. triangular gnomonic numbers)
2 Square numbers
3 Cube numbers
4 Tesseract numbers
5 Penteract numbers
6 Hexeract numbers
7 Hepteract numbers
8 Octeract numbers
9 Enneract numbers
10 Dekeract numbers
11 Hendekeract numbers
12 Dodekeract numbers

### Exponentials

When the base $\scriptstyle b \,$ is fixed, the exponentiation operations are considered exponentials (b^n or b**n)

$b^n \,$

#### Table of exponentials

A sequence of integers $\scriptstyle \{ b^n \}_{n=0}^{\infty} \,$ is called "the exponentials base $\scriptstyle b \,$." Some sequences of exponentials in the OEIS are given in the following table

Table of exponentials
$\scriptstyle b \,$ $\scriptstyle b^n,\, n \,\ge\, 0, \,$ sequences A-number
0[1] {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...} A000007$\scriptstyle (n) \,$
1 {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...} A000012$\scriptstyle (n) \,$
2 {1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, ...} A000079$\scriptstyle (n) \,$
3 {1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, ...} A000244$\scriptstyle (n) \,$
4 {1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304, 16777216, 67108864, 268435456, 1073741824, 4294967296, ...} A000302$\scriptstyle (n) \,$
5 {1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125, 6103515625, 30517578125, ...} A000351$\scriptstyle (n) \,$
6 {1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616, 10077696, 60466176, 362797056, 2176782336, 13060694016, 78364164096, 470184984576, ...} A000400$\scriptstyle (n) \,$
7 {1, 7, 49, 343, 2401, 16807, 117649, 823543, 5764801, 40353607, 282475249, 1977326743, 13841287201, 96889010407, 678223072849, ...} A000420$\scriptstyle (n) \,$
8 {1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 134217728, 1073741824, 8589934592, 68719476736, 549755813888, 4398046511104, ...} A001018$\scriptstyle (n) \,$
9 {1, 9, 81, 729, 6561, 59049, 531441, 4782969, 43046721, 387420489, 3486784401, 31381059609, 282429536481, 2541865828329, 22876792454961, ...} A001019$\scriptstyle (n) \,$
10 {1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, ...} A011557$\scriptstyle (n) \,$
11 {1, 11, 121, 1331, 14641, 161051, 1771561, 19487171, 214358881, 2357947691, 25937424601, 285311670611, 3138428376721, 34522712143931, ...} A001020$\scriptstyle (n) \,$
12 {1, 12, 144, 1728, 20736, 248832, 2985984, 35831808, 429981696, 5159780352, 61917364224, 743008370688, 8916100448256, 106993205379072, ...} A001021$\scriptstyle (n) \,$

#### Exponentials as figurate numbers

The exponentials may be interpreted as the regular orthotopic numbers read cross-dimensionally, although there is a disagreement about 0^0,[1] between the figurate number interpretation (which has to be 0 for $\scriptstyle n \,=\, 0 \,$) and the exponentiation interpretation (which is 1.)

#### Exponentials as a sum of multinomial coefficients

For any positive integer $\scriptstyle m \,$ and any nonnegative integer $\scriptstyle n \,$, the multinomial formula tells us how a polynomial expands when raised to an arbitrary power

$\bigg(\sum_{i=1}^{m} x_i\bigg)^n = \sum_{k_1,k_2,\ldots,k_m} {n \choose k_1, k_2, \ldots, k_m} x_1^{k_1} x_2^{k_2} \cdots x_m^{k_m}, \,$

where

${n \choose k_1, k_2, \ldots, k_m} = {{k_1+k_2+ \ldots +k_m} \choose k_1, k_2, \ldots, k_m} = (k_1,k_2,\ldots,k_m)! = \frac{(k_1+k_2+ \ldots +k_m)!}{k_1!k_2! \cdots k_m!}\,$ are the multinomial coefficients.[2]

Letting all the $\scriptstyle x_i$ equal 1, we get

$\bigg(\sum_{i=1}^{m} 1\bigg)^n = m^n = \sum_{k_1,k_2,\ldots,k_m} {n \choose k_1, k_2, \ldots, k_m}.$

Thus:

$b^n = \sum_{k_1,k_2,\ldots,k_b} {n \choose k_1, k_2, \ldots, k_b}.$

#### Recurrence relation for exponentials

$b^n = b\ b^{n-1}\,$

#### Generating function for exponentials

Since $\scriptstyle \sum_{i=0}^\infty 1 x^i = \frac{1}{1-x},\ |x|<1\,$, the generating function of 1 is then[3] [4]

$G_{\{1^n\}}(x) = G_{\{1\}}(x) = \frac{1}{1-x} = \sum_{n=0}^{\infty} 1^n x^n. \,$

Substituting $\scriptstyle bx \,$ for $\scriptstyle x \,$, we get

$G_{\{b^n\}}(x) = G_{\{1^n\}}(bx) = \frac{1}{1-bx} = \sum_{n=0}^{\infty} b^n x^n, \,$

which is thus the generating function for exponentials.

Setting $\scriptstyle b \,=\, 0 \,$ gives

$G_{\{0^n\}}(x) = G_{\{1^n\}}(0x) = \frac{1}{1-0x} = 1,\ n \ge 0,\,$

which generates the desired sequence for $\scriptstyle 0^n \,$

{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...}

#### Order of basis of exponentials

Any $\scriptstyle k \,\in\, \mathbb{N} \,$ can be uniquely represented, i.e. a representation exists and it is unique, as a sum of powers of a base $\scriptstyle b \,$, i.e.

$k = \sum_{i=0}^{\lfloor\log_{b}(k)\rfloor} d_i b^i,\ d_i \in \{0, 1, \dots, b-2, b-1\}\,$

where the $\scriptstyle d_i \,$ are the digits (i.e. multipliers, or repeated additions, of powers of $\scriptstyle b \,$) of the base $\scriptstyle b \,$ representation.

This is a consequence of the fact that

$\frac{b^n-1}{b-1} = \sum_{i=0}^{n-1} b^i, \,$

or equivalently

$b^n = (b-1) \bigg(\sum_{i=0}^{n-1} b^i\bigg) + 1 = \bigg(\sum_{i=0}^{n-1} (b-1) b^i\bigg) + 1, \,$

which says that $\scriptstyle b^n \,$ is the successor of $\scriptstyle \big(\sum_{i=0}^{n-1} (b-1)\ b^i\big) \,$, where all the digits reached their maximal allowed values.

The number of powers to add reaches local maxima when $\scriptstyle k \,$ is of the form $\scriptstyle b^n - 1 \,$, i.e. we need to add $\scriptstyle (b-1) (1 + \lfloor\log_{b}(k)\rfloor) \,=\, (b-1)n \,$ powers of $\scriptstyle b \,$.

The order of basis of powers $\scriptstyle b^n \,$ is thus infinite, since to represent any $\scriptstyle k \,\in\, \mathbb{N} \,$ we need to add $\scriptstyle O(\log(k)) \,$ powers of $\scriptstyle b \,$.

#### Differences of exponentials

$b^n - b^{n-1} = (b-1)\ b^{n-1}\,$

#### Partial sums of exponentials

$\sum_{n=0}^{m} b^n = \frac{b^{m+1}-1}{b-1},\ b > 1,\,$
$\sum_{n=0}^{m} b^n = m+1,\ b = 1.$

#### Partial sums of reciprocals of exponentials

$\sum_{n=0}^{m} \frac{1}{b^n} = \frac{1}{b^{m}}\ {\sum_{n=0}^{m} b^{m-n}} = \frac{1}{b^{m}}\ {\sum_{n=0}^{m} b^n} = \frac{1}{b^{m}}\ \bigg(\frac{b^{m+1}-1}{b-1}\bigg),\ b > 1,\,$
$\sum_{n=0}^{m} \frac{1}{b^n} = m+1,\ b = 1.$

#### Sum of reciprocals of exponentials

$\sum_{n=0}^{\infty} \frac{1}{b^n} = \sum_{n=0}^{\infty} \bigg(\frac{1}{b}\bigg)^n = \frac{1}{1-\tfrac{1}{b}} = \frac{b}{b-1},\ b > 1,\,$
$\lim_{m \to \infty} \sum_{n=0}^{m} \frac{1}{b^n} = O(m) \to \infty,\ b = 1.$

### N^n

When the base $\scriptstyle n \,$ is equal to the exponent $\scriptstyle n \,$ we get n^n (or n**n), i.e.

$n {\,\uparrow\,} n, \,$

A000312 n^n: number of labeled mappings from $\scriptstyle n \,$ points to themselves (endofunctions), $\scriptstyle n \,\ge\, 0$. (For $\scriptstyle n \,=\, 0$ we get 1 mapping, the empty mapping.)

{1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000, 285311670611, 8916100448256, 302875106592253, 11112006825558016, 437893890380859375, ...}

For example, with n = 5

0 0 0 0 0
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
4 4 4 4 4

one labeled mapping is (0, 0, 2, 3, 0), among $\scriptstyle 5^5$ of them.

## Exponentiation inverses

There are two distinct exponentiation inverses, root extraction and logarithm.

### Root extractions

The $\scriptstyle d \,$th root of $\scriptstyle b^d \,$ is $\scriptstyle b \,$. Root extraction is exponentiation with multiplicative inverse of second term (the exponent, which is the multiplicative inverse of the root index)

$\sqrt[d]{b^d} = (b^d)^{\tfrac{1}{d}} = b \,$

### Logarithms

The logarithm base $\scriptstyle b \,$ of $\scriptstyle b^d \,$ is $\scriptstyle d \,$

$\log_b{b^d} = d \,$

### N//n

When the base $\scriptstyle n \,$ is equal to the root index $\scriptstyle n \,$ we get n//n (inverse operation of n**n,) i.e. n^(1/n)

$\sqrt[n]{n} = n^{\frac{1}{n}} \,$

## Iterated exponentiation

Iterated exponentiation could be abbreviated by the use of a power tower operator (tentatively denoted with the capital letter epsilon of the Greek alphabet,) i.e.

$\underset{i=1}{\overset{n}{\rm E}} b_i = {b_n}^{{.}^{{.}^{{.}^{{b_4}^{{b_3}^{{b_2}^{b_1}}}}}}} \,$

where it should be noted that the power tower is to be evaluated top down.

### Tetration

The particular case (using Knuth's up-arrow notation)

$b{\uparrow \uparrow}d \equiv \underset{i=1}{\overset{d}{\rm E}} b = {b}^{{.}^{{.}^{{.}^{{b}^{{b}^{{b}^{b}}}}}}},\quad d \ge 0, \,$

where for $\scriptstyle d \,=\, 0 \,$ we get the empty tower (actually the empty product, giving the multiplicative identity, i.e. 1,) is called tetration.

It has been attempted to generalize tetration to heights other than nonnegative integers (up to complex numbers.) Some aspects of its formal approach as well as some constants have relations into the integer-sequence-space.

As is the case with exponentiation, we may distinguish between tetra-powers (where the tower height $\scriptstyle d \,$ is fixed)

$n{\uparrow \uparrow}d = \underset{i=1}{\overset{d}{\rm E}} n = {n}^{{.}^{{.}^{{.}^{{n}^{{n}^{{n}^{n}}}}}}},\quad d \ge 0, \,$

and tetra-exponentials (where the tower base $\scriptstyle b \,$ is fixed)

$b{\uparrow \uparrow}n = \underset{i=1}{\overset{n}{\rm E}} b = {b}^{{.}^{{.}^{{.}^{{b}^{{b}^{{b}^{b}}}}}}},\quad n \ge 0. \,$

## Exponentiation identities

### Power identity

The power identity is 1, since $\scriptstyle n^1 \,=\, n \,$ for all $\scriptstyle n \,$.

### Exponential identity

There is no such thing as an exponential identity, since there is no base $\scriptstyle b \,$ such that $\scriptstyle b^n \,=\, n \,$, for all $\scriptstyle n \,$.

## Exponentiation and fixed integer base positional numeral systems

The concept of exponentiation is crucial to our modern place-value systems of numeration; indeed it is the combination of exponentiation (with fixed integer base $\scriptstyle b,\, b \,\ge\, 2 \,$) and addition that represents the advantage of the binary numeral system and the decimal numeral system over non place-value systems of numeration such as Greek numerals, Roman numerals, etc. When in decimal we say "1729," we're in fact saying $\scriptstyle 1 \times 10^3 + 7 \times 10^2 + 2 \times 10^1 + 9 \times 10^0 \,$. Since the exponents for the base $\scriptstyle b \,$ can get arbitrarily large simply by adding more places, there is no need to invent more than $\scriptstyle b \,$ symbols as needs to be done with ancient additive systems.

#### Hierarchical list of operations pertaining to numbers [5] [6]

##### 1st iteration
• Addition:  S(S(… "a times" … (S(n))))
, the sum  n + a
, where  n
is the augend and  a
is the addend. (When addition is commutative both are simply called terms.)
• Subtraction:  P(P(… "s times" … (P(n))))
, the difference  n − s
, where  n
is the minuend and  s
is the subtrahend.
##### 2nd iteration
• Multiplication:  n + (n + (… "k times" … (n + (n))))
, the product  m ⋅ k
, where  m
is the multiplicand and  k
is the multiplier.[7] (When multiplication is commutative both are simply called factors.)
• Division: the ratio  n / d
, where  n
is the dividend and  d
is the divisor.
##### 3rd iteration
• Exponentiation ( d
as "degree",  b
as "base",  n
as "variable").
• Powers:  n ⋅ (n ⋅ (… "d times" … (n ⋅ (n))))
, written  n d
.
• Exponentials:  b ⋅ (b ⋅ (… "n times" … (b ⋅ (b))))
, written  b n
.
• Exponentiation inverses ( d
as "degree",  b
as "base",  n
as "variable").
##### 5th iteration
• Pentation ( d
as "degree",  b
as "base",  n
as "variable").
• Penta-powers:  n ^^ (n ^^ (… "d times" … (n ^^ (n ^^ (n)))))
, written  n ^^^ d or n ↑↑↑ d
.
• Penta-exponentials:  b ^^ (b ^^ (… "n times" … (b ^^ (b ^^ (b)))))
, written  b ^^^ n or b ↑↑↑ n
.
• Pentation inverses
##### 6th iteration
• Hexation ( d
as "degree",  b
as "base",  n
as "variable").
• Hexa-powers:  n ^^^ (n ^^^ (… "d times" … (n ^^^ (n))))
, written  n ^^^^ d or n ↑↑↑↑ d
.
• Hexa-exponentials:  b ^^^ (b ^^^ (… "n times" … (b ^^^ (b))))
, written  b ^^^^ n or b ↑↑↑↑ n
.
• Hexation inverses
##### 7th iteration
• Heptation ( d
as "degree",  b
as "base",  n
as "variable").
• Hepta-powers:  n ^^^^ (n ^^^^ (… "d times" … (n ^^^^ (n))))
, written  n ^^^^^ d or n ↑↑↑↑↑ d
.
• Hepta-exponentials:  b ^^^^ (b ^^^^ (… "n times" … (b ^^^^ (b))))
, written  b ^^^^^ n or b ↑↑↑↑↑ n
.
• Heptation inverses
##### 8th iteration
• Octation ( d
as "degree",  b
as "base",  n
as "variable").
• Octa-powers:  n ^^^^^ (n ^^^^^ (… "d times" … (n ^^^^^ (n))))
, written  n ^^^^^^ d or n ↑↑↑↑↑↑ d
.
• Octa-exponentials:  b ^^^^^ (b ^^^^^ (… "n times" … (b ^^^^^ (b))))
, written  b ^^^^^^ n or b ↑↑↑↑↑↑ n
.
• Octation inverses

## Notes

1. 1.0 1.1 1.2 1.3 Cf. 0^0 or The special case of zero to the zeroeth power.
2. Weisstein, Eric W., Multinomial Coefficient, From MathWorld--A Wolfram Web Resource.
3. Since the power series associated with generating functions are only formal, i.e. used as placeholders for the $\scriptstyle a_n\,$ as coefficients of $\scriptstyle x^n \,$, we need not worry about convergence (as long as it converges for some range of $\scriptstyle x\,$, whatever that range.)
4. Herbert S. Wilf, generatingfunctionology, 2nd ed., 1994.
5. HyperoperationWikipedia.org.
6. Grzegorczyk hierarchyWikipedia.org.
7. There is a lack of consensus on which comes first. Having the multiplier come second makes it consistent with the definitions for exponentiation and higher operations. This is also the convention used with transfinite ordinals:
 ω × 2 := ω + ω
.
Operator precedence

Parenthesization — FactorialExponentiationMultiplication and divisionAddition and subtraction