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Exponentiation with an integer exponent is repetitive multiplication (a 3^{rd} iteration "hyperaddition"): a given number (called the base) is repeatedly multiplied by itself a number of times (called the exponent); this is usually notated and read " exponent ." For example, .
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 "hypermultiplication."
You may also use Knuth's uparrow notation to represent exponentiation.
Exponentiation table
The columns of the table, with fixed exponent , are powers . The rows of the table, with fixed base , are exponentials . The diagonal of the table (entries in bold) are
Exponentiation table
\
 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
Base
Cf. exponentials, exponentiation and fixed integer base positional numeral systems and logarithms.
Exponent
When is positive and is negative, the exponentiations are the reciprocals of the exponentiations of with exponent .
For example, the exponentials (base 2) give .
We have the following rules
 with is .
 for any real, imaginary or complex (including if is interpreted as the empty product, e.g. 1.)

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

we need the conventions

for the constant term to be 1 for any value of , including .
In regards to , see 0^0 or the special case of zero to the zeroeth power.
Powers
When the exponent is fixed, the exponentiation operations are considered powers (n^d or n**d)

Table of powers
A sequence of integers is called "the powers to the degree ." Some sequences of powers in the OEIS are given in the following table
Table of powers
 sequences
 Anumber

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

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

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

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

4
 {0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, ...}
 A000583

5
 {0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, 161051, 248832, 371293, 537824, 759375, 1048576, 1419857, 1889568, 2476099, ...}
 A000584

6
 {0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 16777216, 24137569, ...}
 A001014

7
 {0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 4782969, 10000000, 19487171, 35831808, 62748517, 105413504, 170859375, 268435456, ...}
 A001015

8
 {0, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721, 100000000, 214358881, 429981696, 815730721, 1475789056, 2562890625, ...}
 A001016

9
 {0, 1, 512, 19683, 262144, 1953125, 10077696, 40353607, 134217728, 387420489, 1000000000, 2357947691, 5159780352, 10604499373, 20661046784, ...}
 A001017

10
 {0, 1, 1024, 59049, 1048576, 9765625, 60466176, 282475249, 1073741824, 3486784401, 10000000000, 25937424601, 61917364224, 137858491849, ...}
 A008454

11
 {0, 1, 2048, 177147, 4194304, 48828125, 362797056, 1977326743, 8589934592, 31381059609, 100000000000, 285311670611, 743008370688, ...}
 A008455

12
 {0, 1, 4096, 531441, 16777216, 244140625, 2176782336, 13841287201, 68719476736, 282429536481, 1000000000000, 3138428376721, 8916100448256, ...}
 A008456

Powers as figurate numbers
Powers may be considered as dimensional regular orthotopic 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 is fixed, the exponentiation operations are considered exponentials (b^n or b**n)

Table of exponentials
A sequence of integers is called "the exponentials base ." Some sequences of exponentials in the OEIS are given in the following table
Table of exponentials
 sequences
 Anumber

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

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

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

3
 {1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, ...}
 A000244

4
 {1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304, 16777216, 67108864, 268435456, 1073741824, 4294967296, ...}
 A000302

5
 {1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125, 6103515625, 30517578125, ...}
 A000351

6
 {1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616, 10077696, 60466176, 362797056, 2176782336, 13060694016, 78364164096, 470184984576, ...}
 A000400

7
 {1, 7, 49, 343, 2401, 16807, 117649, 823543, 5764801, 40353607, 282475249, 1977326743, 13841287201, 96889010407, 678223072849, ...}
 A000420

8
 {1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 134217728, 1073741824, 8589934592, 68719476736, 549755813888, 4398046511104, ...}
 A001018

9
 {1, 9, 81, 729, 6561, 59049, 531441, 4782969, 43046721, 387420489, 3486784401, 31381059609, 282429536481, 2541865828329, 22876792454961, ...}
 A001019

10
 {1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, ...}
 A011557

11
 {1, 11, 121, 1331, 14641, 161051, 1771561, 19487171, 214358881, 2357947691, 25937424601, 285311670611, 3138428376721, 34522712143931, ...}
 A001020

12
 {1, 12, 144, 1728, 20736, 248832, 2985984, 35831808, 429981696, 5159780352, 61917364224, 743008370688, 8916100448256, 106993205379072, ...}
 A001021

Exponentials as figurate numbers
The exponentials may be interpreted as the regular orthotopic numbers read crossdimensionally, although there is a disagreement about 0^0,^{[1]} between the figurate number interpretation (which has to be 0 for ) and the exponentiation interpretation (which is 1.)
Exponentials as a sum of multinomial coefficients
For any positive integer and any nonnegative integer , the multinomial formula tells us how a polynomial expands when raised to an arbitrary power

where
 are the multinomial coefficients.^{[2]}
Letting all the equal 1, we get

Thus:

Recurrence relation for exponentials

Generating function for exponentials
Since , the generating function of 1 is then^{[3]} ^{[4]}

Substituting for , we get

which is thus the generating function for exponentials.
Setting gives

which generates the desired sequence for
 {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 can be uniquely represented, i.e. a representation exists and it is unique, as a sum of powers of a base , i.e.

where the are the digits (i.e. multipliers, or repeated additions, of powers of ) of the base representation.
This is a consequence of the fact that

or equivalently

which says that is the successor of , where all the digits reached their maximal allowed values.
The number of powers to add reaches local maxima when is of the form , i.e. we need to add powers of .
The order of basis of powers is thus infinite, since to represent any we need to add powers of .
Differences of exponentials

Partial sums of exponentials


Partial sums of reciprocals of exponentials


Sum of reciprocals of exponentials


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

using Knuth's uparrow notation.
A000312 n^n: number of labeled mappings from points to themselves (endofunctions), . (For 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 of them.
Exponentiation inverses
There are two distinct exponentiation inverses, root extraction and logarithm.
Root extractions
The ^{th} root of is . Root extraction is exponentiation with multiplicative inverse of second term (the exponent, which is the multiplicative inverse of the root index)

Logarithms
The logarithm base of is

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

where it should be noted that the power tower is to be evaluated top down.
Tetration
The particular case (using Knuth's uparrow notation)

where for 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 integersequencespace.
As is the case with exponentiation, we may distinguish between tetrapowers (where the tower height is fixed)

and tetraexponentials (where the tower base is fixed)

Exponentiation identities
Power identity
The power identity is 1, since for all .
Exponential identity
There is no such thing as an exponential identity, since there is no base such that , for all .
Exponentiation and fixed integer base positional numeral systems
The concept of exponentiation is crucial to our modern placevalue systems of numeration; indeed it is the combination of exponentiation (with fixed integer base ) and addition that represents the advantage of the binary numeral system and the decimal numeral system over non placevalue systems of numeration such as Greek numerals, Roman numerals, etc. When in decimal we say "1729," we're in fact saying . Since the exponents for the base can get arbitrarily large simply by adding more places, there is no need to invent more than symbols as needs to be done with ancient additive systems.
See also
Hierarchical list of operations pertaining to numbers ^{[5]} ^{[6]}
0^{th} iteration
1^{st} iteration
 Addition:
S(S(… "a times" … (S(n)))) 
, the sum , where is the augend and is the addend. (When addition is commutative both are simply called terms.)
 Subtraction:
P(P(… "s times" … (P(n)))) 
, the difference , where is the minuend and is the subtrahend.
2^{nd} iteration
 Multiplication:
n + (n + (… "k times" … (n + (n)))) 
, the product , where is the multiplicand and is the multiplier.^{[7]} (When multiplication is commutative both are simply called factors.)
 Division: the ratio , where is the dividend and is the divisor.
3^{rd} iteration
 Exponentiation ( as "degree", as "base", as "variable").
 Powers:
n ⋅ (n ⋅ (… "d times" … (n ⋅ (n)))) 
, written .
 Exponentials:
b ⋅ (b ⋅ (… "n times" … (b ⋅ (b)))) 
, written .
 Exponentiation inverses ( as "degree", as "base", as "variable").
4^{th} iteration
 Tetration ( as "degree", as "base", as "variable").
 Tetration inverses ( as "degree", as "base", as "variable").
5^{th} iteration
 Pentation ( as "degree", as "base", as "variable").
 Pentation inverses
6^{th} iteration
 Hexation ( as "degree", as "base", as "variable").
 Hexation inverses
7^{th} iteration
 Heptation ( as "degree", as "base", as "variable").
 Heptation inverses
8^{th} iteration
 Octation ( as "degree", as "base", as "variable").
 Octapowers:
n ^^^^^ (n ^^^^^ (… "d times" … (n ^^^^^ (n)))) 
, written .
 Octaexponentials:
b ^^^^^ (b ^^^^^ (… "n times" … (b ^^^^^ (b)))) 
, written .
 Octation inverses
Notes
 ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} Cf. 0^0 or The special case of zero to the zeroeth power.
 ↑ Weisstein, Eric W., Multinomial Coefficient, From MathWorldA Wolfram Web Resource.
 ↑ Since the power series associated with generating functions are only formal, i.e. used as placeholders for the as coefficients of , we need not worry about convergence (as long as it converges for some range of , whatever that range.)
 ↑ Herbert S. Wilf, generatingfunctionology, 2^{nd} ed., 1994.
 ↑ Hyperoperation—Wikipedia.org.
 ↑ Grzegorczyk hierarchy—Wikipedia.org.
 ↑ 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: .
Notes