Exponentiation with an integer exponent Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle d \,}
is repetitive multiplication (a 3rd iteration "hyper-addition"): a given number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b \,}
(called the base) is repeatedly multiplied by itself a number of times Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle d \,}
(called the exponent); this is usually notated Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b^d \,}
and read "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b \,}
exponent Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle d \,}
." For example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b{\uparrow}d \,}
to represent exponentiation.
Exponentiation table
The columns of the table, with fixed exponent Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle d \,}
, are powers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle n^d \,}
. The rows of the table, with fixed base Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b \,}
, are exponentials Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b^n \,}
. The diagonal of the table (entries in bold) are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle n^n \,}
Exponentiation table Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b^d\, (b \,\ge\, 0,\, d \,\ge\, 0) \,}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle b \,}
\ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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
Base
Cf. exponentials, exponentiation and fixed integer base positional numeral systems and logarithms.
Exponent
When Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b \,}
is positive and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle d \,}
is negative, the exponentiations are the reciprocals of the exponentiations of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b \,}
with exponent Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle |d| \,}
.
For example, the exponentials (base 2) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \{2^{-i}\}_{i=0}^{\infty} \,}
give Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \big\{1, \tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{8}, \cdots\big\} \,}
.
We have the following rules
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b^d \,}
with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle d \,<\, 0 \,}
is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \frac{1}{b^{|d|}},\ b \,\neq\, 0 \,}
.
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b^0 \,=\, 1 \,}
for any real, imaginary or complex Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b \,}
(including Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b \,=\, 0 \,}
if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b^0 \,}
is interpreted as the empty product, e.g. 1.)
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b^1 \,=\, b \,}
0^0
If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle n \,}
, including 0.
In algebra, for the binomial expansion
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (1+x)^n = \sum_{i=0}^{n} \binom{n}{i} x^i \,}
we need the conventions
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0! = 1,\ 0^0 = 1 \,}
for the constant term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \binom{n}{0} x^0 \,=\, \frac{n!}{n!0!} x^0 \,}
to be 1 for any value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle x \,}
, including Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle x \,=\, 0 \,}
.
In regards to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle 0^0 \,}
, see 0^0 or the special case of zero to the zeroeth power.
Powers
When the exponent Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle d \,}
is fixed, the exponentiation operations are considered powers (n^d or n**d)
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n^d \,}
Table of powers
A sequence of integers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \{ n^d \}_{n=0}^{\infty} \,}
is called "the powers to the degree Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle d \,}
." Some sequences of powers in the OEIS are given in the following table
Table of powers
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle d \,}
|
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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, ...}
|
A000012Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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, ...}
|
A000027Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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, ...}
|
A000290Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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, ...}
|
A000578Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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, ...}
|
A000583Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (n) \,}
|
5
|
{0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, 161051, 248832, 371293, 537824, 759375, 1048576, 1419857, 1889568, 2476099, ...}
|
A000584Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (n) \,}
|
6
|
{0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 16777216, 24137569, ...}
|
A001014Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (n) \,}
|
7
|
{0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 4782969, 10000000, 19487171, 35831808, 62748517, 105413504, 170859375, 268435456, ...}
|
A001015Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (n) \,}
|
8
|
{0, 1, 256, 6561, 65536, 390625, 1679616, 5764801, 16777216, 43046721, 100000000, 214358881, 429981696, 815730721, 1475789056, 2562890625, ...}
|
A001016Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (n) \,}
|
9
|
{0, 1, 512, 19683, 262144, 1953125, 10077696, 40353607, 134217728, 387420489, 1000000000, 2357947691, 5159780352, 10604499373, 20661046784, ...}
|
A001017Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (n) \,}
|
10
|
{0, 1, 1024, 59049, 1048576, 9765625, 60466176, 282475249, 1073741824, 3486784401, 10000000000, 25937424601, 61917364224, 137858491849, ...}
|
A008454Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (n) \,}
|
11
|
{0, 1, 2048, 177147, 4194304, 48828125, 362797056, 1977326743, 8589934592, 31381059609, 100000000000, 285311670611, 743008370688, ...}
|
A008455Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (n) \,}
|
12
|
{0, 1, 4096, 531441, 16777216, 244140625, 2176782336, 13841287201, 68719476736, 282429536481, 1000000000000, 3138428376721, 8916100448256, ...}
|
A008456Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (n) \,}
|
Powers as figurate numbers
Powers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle n^d \,}
may be considered as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle d \,}
-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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b \,}
is fixed, the exponentiation operations are considered exponentials (b^n or b**n)
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle b^n \,}
Table of exponentials
A sequence of integers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \{ b^n \}_{n=0}^{\infty} \,}
is called "the exponentials base Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b \,}
." Some sequences of exponentials in the OEIS are given in the following table
Table of exponentials
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b \,}
|
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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, ...}
|
A000007Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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, ...}
|
A000012Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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, ...}
|
A000079Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (n) \,}
|
3
|
{1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, ...}
|
A000244Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (n) \,}
|
4
|
{1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304, 16777216, 67108864, 268435456, 1073741824, 4294967296, ...}
|
A000302Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (n) \,}
|
5
|
{1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125, 6103515625, 30517578125, ...}
|
A000351Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (n) \,}
|
6
|
{1, 6, 36, 216, 1296, 7776, 46656, 279936, 1679616, 10077696, 60466176, 362797056, 2176782336, 13060694016, 78364164096, 470184984576, ...}
|
A000400Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (n) \,}
|
7
|
{1, 7, 49, 343, 2401, 16807, 117649, 823543, 5764801, 40353607, 282475249, 1977326743, 13841287201, 96889010407, 678223072849, ...}
|
A000420Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (n) \,}
|
8
|
{1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 134217728, 1073741824, 8589934592, 68719476736, 549755813888, 4398046511104, ...}
|
A001018Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (n) \,}
|
9
|
{1, 9, 81, 729, 6561, 59049, 531441, 4782969, 43046721, 387420489, 3486784401, 31381059609, 282429536481, 2541865828329, 22876792454961, ...}
|
A001019Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (n) \,}
|
10
|
{1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, ...}
|
A011557Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (n) \,}
|
11
|
{1, 11, 121, 1331, 14641, 161051, 1771561, 19487171, 214358881, 2357947691, 25937424601, 285311670611, 3138428376721, 34522712143931, ...}
|
A001020Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (n) \,}
|
12
|
{1, 12, 144, 1728, 20736, 248832, 2985984, 35831808, 429981696, 5159780352, 61917364224, 743008370688, 8916100448256, 106993205379072, ...}
|
A001021Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle n \,=\, 0 \,}
) and the exponentiation interpretation (which is 1.)
Exponentials as a sum of multinomial coefficients
For any positive integer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle m \,}
and any nonnegative integer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle n \,}
, the multinomial formula tells us how a polynomial expands when raised to an arbitrary power
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle x_i}
equal 1, we get
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle b^n = \sum_{k_1,k_2,\ldots,k_b} {n \choose k_1, k_2, \ldots, k_b}.}
Recurrence relation for exponentials
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle b^n = b\ b^{n-1}\,}
Generating function for exponentials
Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \sum_{i=0}^\infty 1 x^i = \frac{1}{1-x},\ |x|<1\,}
, the generating function of 1 is then[3] [4]
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle G_{\{1^n\}}(x) = G_{\{1\}}(x) = \frac{1}{1-x} = \sum_{n=0}^{\infty} 1^n x^n. \,}
Substituting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle bx \,}
for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle x \,}
, we get
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b \,=\, 0 \,}
gives
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle G_{\{0^n\}}(x) = G_{\{1^n\}}(0x) = \frac{1}{1-0x} = 1,\ n \ge 0,\,}
which generates the desired sequence for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \scriptstyle b \,}
, i.e.
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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
are the digits (i.e. multipliers, or repeated additions, of powers of
) of the base
representation.
This is a consequence of the fact that
![{\displaystyle {\frac {b^{n}-1}{b-1}}=\sum _{i=0}^{n-1}b^{i},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/0d0c1f2b57a4f4ba73d1519fbdb8edb20b23c3d8)
or equivalently
![{\displaystyle 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,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/8de2c6a2811eb3e9563c366e51b761551f5825b9)
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
![{\displaystyle b^{n}-b^{n-1}=(b-1)\ b^{n-1}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/70504919326258e68d7164f696d7f5012ab80cc8)
Partial sums of exponentials
![{\displaystyle \sum _{n=0}^{m}b^{n}={\frac {b^{m+1}-1}{b-1}},\ b>1,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/b76b4c90e1a532b117f8991eb5b1f218de8a2856)
![{\displaystyle \sum _{n=0}^{m}b^{n}=m+1,\ b=1.}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/a28e79f6577c2d393c4345b5810ea90326c2661a)
Partial sums of reciprocals of exponentials
![{\displaystyle \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,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/b793722e6710d94a4c1c338784fdbce5c7ef7f28)
Sum of reciprocals of exponentials
![{\displaystyle \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,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/80772460c594d12062f334f898349e0581ac45f9)
![{\displaystyle \lim _{m\to \infty }\sum _{n=0}^{m}{\frac {1}{b^{n}}}=O(m)\to \infty ,\ b=1.}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/b7ee2dc76f3326c091d96c1387a2b31ba0ad1fbc)
N^n
When the base
is equal to the exponent
we get n^n (or n**n), i.e.
![{\displaystyle n{\,\uparrow \,}n,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/8ebc6e7acddade4f83eb94033559a754151f11aa)
using Knuth's up-arrow 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
- 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.
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)
![{\displaystyle {\sqrt[{d}]{b^{d}}}=(b^{d})^{\tfrac {1}{d}}=b\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/3424f51af11b79197a4d136fc9c70ee28fec67a0)
Logarithms
The logarithm base
of
is
![{\displaystyle \log _{b}{b^{d}}=d\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e64e00c58b36080f4b0f4f3e6dfb573aeacd26e7)
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)
![{\displaystyle {\sqrt[{n}]{n}}=n^{\frac {1}{n}}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/50810dbb45fc68734514dfc385164da5e15d8755)
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.
![{\displaystyle {\underset {i=1}{\overset {n}{\rm {E}}}}b_{i}={b_{n}}^{{.}^{{.}^{{.}^{{b_{4}}^{{b_{3}}^{{b_{2}}^{b_{1}}}}}}}}\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/aee038348987055617ace50dab9d1e29ff183976)
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)
![{\displaystyle b{\uparrow \uparrow }d\equiv {\underset {i=1}{\overset {d}{\rm {E}}}}b={b}^{{.}^{{.}^{{.}^{{b}^{{b}^{{b}^{b}}}}}}},\quad d\geq 0,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/1cfb8d5a452142b4003a425dced21f0bed3b1b49)
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 integer-sequence-space.
As is the case with exponentiation, we may distinguish between tetra-powers (where the tower height
is fixed)
![{\displaystyle n{\uparrow \uparrow }d={\underset {i=1}{\overset {d}{\rm {E}}}}n={n}^{{.}^{{.}^{{.}^{{n}^{{n}^{{n}^{n}}}}}}},\quad d\geq 0,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/447bfed6d041e942ebe97cb3c25a940619fb1eef)
and tetra-exponentials (where the tower base
is fixed)
![{\displaystyle b{\uparrow \uparrow }n={\underset {i=1}{\overset {n}{\rm {E}}}}b={b}^{{.}^{{.}^{{.}^{{b}^{{b}^{{b}^{b}}}}}}},\quad n\geq 0.\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f4f6740173d2c8a0667c5694f5de0b718efb200f)
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 place-value 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 place-value 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]
0th iteration
1st 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.
2nd 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.
3rd 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").
4th iteration
- Tetration ( as "degree", as "base", as "variable").
- Tetration inverses ( as "degree", as "base", as "variable").
5th iteration
- Pentation ( as "degree", as "base", as "variable").
- Pentation inverses
6th iteration
- Hexation ( as "degree", as "base", as "variable").
- Hexation inverses
7th iteration
- Heptation ( as "degree", as "base", as "variable").
- Heptation inverses
8th iteration
- Octation ( as "degree", as "base", as "variable").
- Octa-powers:
n ^^^^^ (n ^^^^^ (⋯ "d times" ⋯ (n ^^^^^ (n)))) |
, written .
- Octa-exponentials:
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 MathWorld--A 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, 2nd 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