Exponentiation

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

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 2^nd iteration "hyper-multiplication."

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

Exponentiation table

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

Exponentiation table $b^d(b\ge 0,d\ge 0)$

$b$ \ $d$	0	1	2	3	4	5	6	7	8	9	10	11	12
0	1	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

Base

Cf. exponentials, exponentiation and fixed integer base positional numeral systems and logarithms.

Exponent

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

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

We have the following rules

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

0^0

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

In algebra, for the binomial expansion

$(1+x{)}^n=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})x^i$

we need the conventions

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

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

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

Powers

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

$n^d$

Table of powers

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

Table of powers

$d$	$n^d,n\ge 0,$ 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, ...}	A000012$(n)$
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$(n)$
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$(n)$
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$(n)$
4	{0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, ...}	A000583$(n)$
5	{0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, 161051, 248832, 371293, 537824, 759375, 1048576, 1419857, 1889568, 2476099, ...}	A000584$(n)$
6	{0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 16777216, 24137569, ...}	A001014$(n)$
7	{0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 4782969, 10000000, 19487171, 35831808, 62748517, 105413504, 170859375, 268435456, ...}	A001015$(n)$
8	{0, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721, 100000000, 214358881, 429981696, 815730721, 1475789056, 2562890625, ...}	A001016$(n)$
9	{0, 1, 512, 19683, 262144, 1953125, 10077696, 40353607, 134217728, 387420489, 1000000000, 2357947691, 5159780352, 10604499373, 20661046784, ...}	A001017$(n)$
10	{0, 1, 1024, 59049, 1048576, 9765625, 60466176, 282475249, 1073741824, 3486784401, 10000000000, 25937424601, 61917364224, 137858491849, ...}	A008454$(n)$
11	{0, 1, 2048, 177147, 4194304, 48828125, 362797056, 1977326743, 8589934592, 31381059609, 100000000000, 285311670611, 743008370688, ...}	A008455$(n)$
12	{0, 1, 4096, 531441, 16777216, 244140625, 2176782336, 13841287201, 68719476736, 282429536481, 1000000000000, 3138428376721, 8916100448256, ...}	A008456$(n)$

Powers as figurate numbers

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

Regular orthotopic numbers

$d$	$d$-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

Formulae

Cf. Formulae for regular orthotopic numbers.

Recurrence relation for powers

Cf. Recurrence relation for regular orthotopic numbers.

Generating function for powers

Cf. Generating function for regular orthotopic numbers.

Order of basis of powers

Cf. Order of basis of regular orthotopic numbers.

Differences of powers

Cf. Differences of regular orthotopic numbers.

Partial sums of powers

Cf. Partial sums of regular orthotopic numbers.

Partial sums of reciprocals of powers

Cf. Partial sums of reciprocals of regular orthotopic numbers.

Sum of reciprocals of powers

Cf. Sum of reciprocals of regular orthotopic numbers.

Exponentials

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

$b^n$

Table of exponentials

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

Table of exponentials

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

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, between the figurate number interpretation (which has to be 0 for $n=0$) and the exponentiation interpretation (which is 1.)

Exponentials as a sum of multinomial coefficients

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

$(\sum _{i=1}^m{x}_i{)}^n=\sum _{k_1,k_2,\dots ,k_m}(\genfrac{}{}{0ex}{}{n}{k_1,k_2,\dots ,k_m})x_1^{k_1}{x}_2^{k_2}\cdots x_m^{k_m},$

where

$(\genfrac{}{}{0ex}{}{n}{k_1,k_2,\dots ,k_m})=(\genfrac{}{}{0ex}{}{k_1+k_2+\dots +k_m}{k_1,k_2,\dots ,k_m})=(k_1,k_2,\dots ,k_m)!=\frac{(k_1+k_2+\dots +k_m)!}{k_1!k_2!\cdots k_m!}$ are the multinomial coefficients.

Letting all the $x_i$ equal 1, we get

$(\sum _{i=1}^{m}1{)}^n=m^n=\sum _{k_1,k_2,\dots ,k_m}(\genfrac{}{}{0ex}{}{n}{k_1,k_2,\dots ,k_m}).$

Thus:

$b^n=\sum _{k_1,k_2,\dots ,k_b}(\genfrac{}{}{0ex}{}{n}{k_1,k_2,\dots ,k_b}).$

Recurrence relation for exponentials

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

Generating function for exponentials

Since ${\sum }_{i=0}^{\mathrm{\infty }}1x^i=\frac{1}{1-x},|x|<1$, the generating function of 1 is then

$G_{\{1^n\}}(x)=G_{\{1\}}(x)=\frac{1}{1-x}=\sum _{n=0}^{\mathrm{\infty }}{1}^n{x}^n.$

Substituting $bx$ for $x$, we get

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

which is thus the generating function for exponentials.

Setting $b=0$ gives

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

which generates the desired sequence for $0^n$

{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 $k\in \mathbb{\mathbb{N}}$ can be uniquely represented, i.e. a representation exists and it is unique, as a sum of powers of a base $b$, 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 $d_i$ are the digits (i.e. multipliers, or repeated additions, of powers of $b$) of the base $b$ representation.

This is a consequence of the fact that

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

or equivalently

$b^n=(b-1)\left(\sum _{i=0}^{n-1}{b}^i\right)+1=(\sum _{i=0}^{n-1}(b-1)b^i)+1,$

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

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

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

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

$\sum _{n=0}^m\frac{1}{b^n}=\frac{1}{b^m}\sum _{n=0}^m{b}^{m-n}=\frac{1}{b^m}\sum _{n=0}^m{b}^n=\frac{1}{b^m}\left(\frac{b^{m+1}-1}{b-1}\right),b>1,$

$\sum _{n=0}^m\frac{1}{b^n}=m+1,b=1.$

Sum of reciprocals of exponentials

$\sum _{n=0}^{\mathrm{\infty }}\frac{1}{b^n}=\sum _{n=0}^{\mathrm{\infty }}(\frac{1}{b}{)}^n=\frac{1}{1-\frac{1}{b}}=\frac{b}{b-1},b>1,$

$\underset{m\to \mathrm{\infty }}{\lim }\sum _{n=0}^m\frac{1}{b^n}=O(m)\to \mathrm{\infty },b=1.$

N^n

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

$n\uparrow n,$

using Knuth's up-arrow notation.

A000312 n^n: number of labeled mappings from $n$ points to themselves (endofunctions), $n\ge 0$. (For $n=0$ 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 $5^5$ of them.

Exponentiation inverses

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

Root extractions

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

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

Logarithms

The logarithm base $b$ of $b^d$ is $d$

${\log }_b{b}^d=d$

N//n

When the base $n$ is equal to the root index $n$ 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.

$\underset{i=1}{\stackrel{n}{\mathrm{E}}}{b}_i={b_n}^{{.}^{{.}^{{.}^{{b_4}^{{b_3}^{{b_2}^{b_1}}}}}}}$

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)

$b\uparrow \uparrow d\equiv \underset{i=1}{\stackrel{d}{\mathrm{E}}}b=b^{{.}^{{.}^{{.}^{b^{b^{b^b}}}}}},d\ge 0,$

where for $d=0$ 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 $d$ is fixed)

$n\uparrow \uparrow d=\underset{i=1}{\stackrel{d}{\mathrm{E}}}n=n^{{.}^{{.}^{{.}^{n^{n^{n^n}}}}}},d\ge 0,$

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

$b\uparrow \uparrow n=\underset{i=1}{\stackrel{n}{\mathrm{E}}}b=b^{{.}^{{.}^{{.}^{b^{b^{b^b}}}}}},n\ge 0.$

Exponentiation identities

Power identity

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

Exponential identity

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

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 $b,b\ge 2$) 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 $1\times 10^3+7\times 10^2+2\times 10^1+9\times 10^0$. Since the exponents for the base $b$ can get arbitrarily large simply by adding more places, there is no need to invent more than $b$ symbols as needs to be done with ancient additive systems.

See also

• Category:Unary operations (numbers)
• Category:Binary operations (numbers)

• Knuth's up-arrow notation
• Conway's chained arrow notation

Hierarchical list of operations pertaining to numbers

0^th iteration

• Successor:  

  S(n)

  .
• Predecessor:  

  P(n)

  .

1^st 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.

2^nd iteration

• Multiplication:  

  n + (n + (⋯ "k times" ⋯ (n + (n))))

  , the product  

  m  ⋅   k

  , where  

  m

  is the multiplicand and  

  k

  is the multiplier. (When multiplication is commutative both are simply called factors.)
• Division: the ratio  

  n  /  d

  , where  

  n

  is the dividend and  

  d

  is the divisor.
  • Quotient: (integer division).
  • Remainder: (modulo and congruences).

3^rd 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

    .
    • Exponential function:  

      e n

      , where  

      e

      is Euler's number.
• Exponentiation inverses ( 

  d

  as "degree",  

  b

  as "base",  

  n

  as "variable").
  • Roots:  

    d √  n

    .
  • Logarithms:  

    logb n

    .
    • Natural logarithm function:  

      log n

      , or  

      loge n

      , where  

      e

      is Euler's number.

4^th iteration

• Tetration ( 

  d

  as "degree",  

  b

  as "base",  

  n

  as "variable").
  • Tetra-powers (super-powers):  

    n ^ (n ^ (⋯ "d times" ⋯ (n ^ (n))))

    , written  

    n ^^ d or n ↑↑ d

    .
  • Tetra-exponentials (super-exponentials):  

    b ^ (b ^ (⋯ "n times" ⋯ (b ^ (b))))

    , written  

    b ^^ n or b ↑↑ n

    .
• Tetration inverses ( 

  d

  as "degree",  

  b

  as "base",  

  n

  as "variable").
  • Tetra-roots (super-roots)
  • Tetra-logarithms (super-logarithms):  

    slogb n

    .
    • Iterated logarithm:  

      log ⁎b n = ⌈slogb n⌉

      .

5^th 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
  • Penta-roots
  • Penta-logarithms

6^th 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
  • Hexa-roots
  • Hexa-logarithms

7^th 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
  • Hepta-roots
  • Hepta-logarithms

8^th 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
  • Octa-roots
  • Octa-logarithms

Notes

Operator precedence
Parenthesization — Factorial — Exponentiation — Multiplication and division — Addition and subtraction