Exponentiation

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Exponentiation is a Mathematical Operation , written '''''a''''n''''', involving two numbers, the ''' Base ''' ''a'' and the '''exponent''' ''n''. When ''n'' is a Positive Integer , exponentiation corresponds to repeated Multiplication : :$a^n\; =\; \backslash underbrace\{a\; imes\; \backslash cdots\; imes\; a\}\_n,$ just as multiplication by a whole number corresponds to repeated Addition : :$a\; imes\; n\; =\; \backslash underbrace\{a\; +\; \backslash cdots\; +\; a\}\_n.$ The exponent is usually shown as a Superscript to the right of the base. The exponentiation ''a''''n'' can be read as: ''a raised to the n-th power'' or ''a raised to the power n'', or more briefly: ''a to the n-th power'' or ''a to the power n''. Some exponents can be read in a certain way; for example ''a''2 is usually read as ''a squared'' and ''a''3 as ''a cubed''. The power ''a''''n'' can also be defined when the exponent ''n'' is a negative integer. When the base ''a'' is a positive real number, exponentiation is defined for real and even complex exponents ''n''. The special Exponential Function ''e''''x'' is fundamental for this definition. It enables the functions of Trigonometry to be expressed by exponentiation. However, when the base ''a'' is not a positive real number and the exponent ''n'' is not an integer, then ''a''''n'' cannot be defined as a unique Continuous Function of ''a''. Exponentiation where the exponent is a Matrix is used for solving systems of Linear Differential Equation s. Exponentiation is used pervasively in many other fields as well, including economics, biology, chemistry, physics, and computer science, with applications such as Compound Interest , Population Growth , chemical Reaction Kinetics , Wave behavior, and Public Key Cryptography . EXPONENTIATION WITH INTEGER EXPONENTS The exponentiation operation with integer exponents only requires Elementary Algebra . Positive integer exponents ''a''2 = ''a''·''a'' is called the Square of ''a'' because the area of a square with side-length ''a'' is ''a''2. ''a''3 = ''a''·''a''·''a'' is called the Cube , because the volume of a cube with side-length ''a'' is ''a''3. So 32 is pronounced "three squared",and 23 is "two cubed". The exponent says how many copies of the base are multiplied together. For example, 35 = 3·3·3·3·3 = 243. The base 3 appears 5 times in the repeated multiplication, because the exponent is 5. Here, 3 is the ''base'', 5 is the ''exponent'', and 243 is the ''power'' or, more specifically, ''the fifth power of 3'' or ''3 raised to the fifth power''. The word "raised" is usually omitted, and most often "power" as well, so 35 is typically pronounced "three to the fifth" or "three to the five". Formally, powers with positive integer exponents may be defined by the initial condition ''a''1 = ''a'' and the Recurrence Relation ''a''''n''+1 = ''a''·''a''''n''. Exponents one and zero Notice that 31 is product of only one 3, which is evidently 3. Also note that 35 = 3·34. Also 34 = 3·33. Continuing this trend, we should have :31 = 3·30 Another way of saying this is that when ''n'', ''m'', and ''n'' - ''m'' are positive (and if ''x'' is not equal to zero), one can see by counting the number of occurrences of ''x'' that : Extended to the case that ''n'' and ''m'' are equal, the equation would read : since both the numerator and the denominator are equal. Therefore we take this as the definition of ''x''0. Therefore we define 30 = 1 so that the above equality holds. This leads to the following rule: Any number to the power 1 is itself. Any nonzero number to the power 0 is 1; one interpretation of these powers is as Empty Product s. The case of 00 is discussed Below . Combinatorial interpretation For non-negative integers ''n'' and ''m'', the power ''n''''m'' equals the Cardinality of the set of ''m''- Tuple s from an ''n''-element Set , or the number of ''m''-letter words from an ''n''-letter alphabet.
	:1<sup>4</sup>	{ (1,1,1,1) } = 1 There is one 4-tuple from a one-element set
	:2<sup>3</sup>	{ (1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2), (2,2,1), (2,2,2) } = 8 There are 8 3-tuples from a two-element set
	:3<sup>2</sup>	{ (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3) } = 9 There are 9 2-tuples from a three-element set
	:4<sup>1</sup>	{ (1), (2), (3), (4) } = 4 There are 4 1-tuples from a four-element set
	:5<sup>0</sup>	{ () } = 1 There is exactly one empty Tuple

Identities and properties

The most important Identity satisfied by integer exponentiation is:
:$a^\{m\; +\; n\}\; =\; a^m\; \backslash cdot\; a^n$
This identity has the consequence:
:
for ''a'' ≠ 0, and
:$(a^m)^n\; =\; a^\{m\; \backslash cdot\; n\}.$

While addition and multiplication are Commutative (for example,
2+3 = 5 = 3+2 and
2·3 = 6 = 3·2), exponentiation is not commutative:
2³ = 8, but 3² = 9.

Similarly, while addition and multiplication are Associative (for example,
(2+3)+4 = 9 = 2+(3+4) and
(2·3)·4 = 24 = 2·(3·4), exponentiation is not associative either:
2³ to the 4th power is 8⁴ or 4096, but 2 to the 3⁴ power is
2⁸¹ or 2,417,851,639,229,258,349,412,352. Without parentheses to modify the order of calculation, the order is usually understood to be from right to left:
:$a^\{b^c\}=a^\{(b^c)\}$
e (a^b)^c=a^{(b\cdot c)}=a^{b\cdot c}


Powers of ten


Powers of 10 are easily computed in the base ten ( Decimal ) number system. For example, 10⁸ = 100000000.

Exponentiation with base 10 is used in Scientific Notation to describe large or small numbers. For instance, 299,792,458 (the Speed Of Light in a vacuum, in meters per second) can be written as 2.99792458·10⁸ and then Approximated as 2.998·10⁸, (or sometimes as 299.8·10⁶, or 299.8E+6, especially in computer software).

SI Prefix es based on powers of 10 are also used to describe small or large quantities. For example, the prefix Kilo means 10³ = 1000, so a kilometre is 1000 Metre s.


Powers of two

The positive Powers Of 2 are important in Computer Science because there are 2^''n'' possible values for an ''n''- Bit Variable . See Binary Numeral System .

Powers of 2 are important in Set Theory since a set with ''n'' members has a Power Set , or set of all Subset s of the original set, with 2^''n'' members.

The negative powers of 2 are commonly used, and the first two have special names: Half , and Quarter .


Powers of zero

If the exponent is positive, the power of zero is zero: 0^''n'' = 0, where ''n'' > 0.

If the exponent is negative, the power of zero (0^−''n'', where ''n'' > 0) remains undefined, because division by zero is implied.

If the exponent is zero, some authors define 0⁰=1, whereas others leave it undefined, as discussed Below .


Powers of minus one

The powers of minus one are useful for expressing alternating sequences.

If the exponent is even, the power of minus one is one: (−1)^2''n'' = 1.

If the exponent is odd, the power of minus one is minus one: (−1)^2''n''+1 = −1.


Powers of the imaginary unit

The powers of the Imaginary Unit ''i'' are useful for expressing sequences of period 4. See for example Root Of Unity#Periodicity .

: $i^\{4n+1\}=i\; \backslash !\backslash$

: $i^\{4n+2\}=-1\; \backslash !\backslash$

: $i^\{4n+3\}=-i\; \backslash !\backslash$

: $i^\{4n\}=1\; \backslash !\backslash$


Powers of e

See Also: Exponential function


The number ''e'' , the base of the Natural Logarithm , is a well studied constant approximately equal to 2.718. The function ''e''^''x'', known as the exponential function, has applications in many areas of mathematics and science.

The function ''e''^''x'' can be computed in many ways, including the Limit
:$$
e^x = \lim_{n ightarrow \infty} \left(1+rac{x}{n} ight) ^n\,

and the Power Series
:.

These formulas for ''e''^''x'' only require that ''x'' can be raised to positive integer powers, and thus the formulas can be used not only for integer values of ''x'', but for fractional, real, or complex values. For this reason, the exponential function can be used to define ''a''^''b'' for general real or complex values of ''a'' and ''b'', as discussed below. A further generalization is the computation of ''e''^''A'' where ''A'' is a Square Matrix , which is used to solve Ordinary Differential Equation s.


POWERS OF REAL NUMBERS

(red), base 2 (blue), base ½ (cyan). Note how all of the curves pass through the point (0, 1). This is because, in accordance with The Properties Of Exponentiation , any non-zero number raised to the power 0 is 1. Also note that at x=1, the y value equals the base. This is because any number raised to the power 1 is that same number.]]

Raising a positive real number to a power that is not an integer can be accomplished in two ways.

• Rational Number exponents can be defined in terms of ''n^th'' Root s, and arbitrary nonzero exponents can then be defined by continuity.


• The Natural Logarithm can be used to define real exponents using the exponential function.

The Identities And Properties shown above are true for non-integer exponents as well.


Principal n-th root

See Also: n-th root



An ''n''th root of a Number ''a'' is a number ''b'' such that ''bⁿ'' = ''a''.

When referring to ''the'' ''n''th root of a Real Number ''a'' it is assumed that what is desired is the principal ''n''th root of the number. If ''a'' is a real number, and ''n'' is a positive integer, then the unique real solution with the same sign as ''a'' to the equation
:$\backslash \; x^n\; =\; a$
is called the principal N^th Root of ''a'', and is denoted $\backslash sqrt${Link without Title} {a} using the radical Symbol $(\backslash sqrt\{\backslash ,\backslash ,\}).$.

:{Link without Title} {a}.

For example: 4^1/2 = 2, 8^1/3 = 2, -8^1/3 = -2, .

Note that if ''n'' is Even , Negative Number s will not have a principal ''n''th root.


Rational powers of positive real numbers


Exponentiation with a Rational exponent ''m''/''n'' can be defined as

:{Link without Title} {a})^m.

For example, 8^2/3 = 4.

Since any Real Number can be approximated by rational numbers, exponentiation to an arbitrary real exponent ''k'' can be defined by Continuity with the rule

:$a^k\; =\; \backslash lim\_\{r\; o\; k\}\; a^r,$
where the limit is taken only over rational values of ''r''.

For example, if
:$k\; \backslash approx\; 1.732$
then
:$5^k\; \backslash approx\; 5^\{1.732\}.$


Real powers of positive real numbers


The Natural Logarithm ln(''x'') is the Inverse of the exponential function ''e''^''x''. It is defined for every positive real number ''b'' and satisfies the equation

:$b\; =\; e^\{\backslash ln\; b\}.\backslash ,$

Assuming ''b''^''x'' is already defined, logarithm and exponent rules give the equality

:$b^x\; =\; (e^\{\backslash ln\; b\})^x\; =\; e^\{x\; \backslash cdot\backslash ln\; b\}.\backslash ,$

This equality can be used to define exponentiation with any positive real base ''b'' as
:$b^x\; =\; e^\{x\backslash cdot\backslash ln\; b\}.\backslash ,$

This definition of the real number power ''b''^''x'' agrees with the definition given above using rational exponents and continuity. The definition of exponentiation using logarithms is more common in the context of complex numbers, as discussed below.


Some rational powers of negative real numbers

Neither the logarithm method nor the fractional exponent method can be used to define ''a''^''k'' as a real number for a negative real number ''a'' and an arbitrary real number ''k''. In some special cases, a definition is possible: integral powers of negative real numbers are real numbers, and rational powers of the form ''a''^''m''/''n'' where ''n'' is odd can be computed using roots. But since there is no real number ''x'' such that ''x''² = −1, the definition of ''a''^''m''/''n'' when ''n'' is even and ''m'' is odd must use the Imaginary Unit ''i'', as described more fully in the next section.

The logarithm method cannot be used to define ''a''^''k'' as a real number when ''a'' < 0 because ''e''^''x'' is nonnegative for every real number ''x'', so log(''a'') cannot be a real number.

The rational exponent method cannot be used for negative values of ''a'' because it relies on Continuity . The function ''f''(''r'') = ''a''^''r'' has a unique continuous extension from the rational numbers to the real numbers for each ''a'' > 0. But when ''a'' < 0, the function ''f'' is not even continuous on the set of rational numbers ''r'' for which it is defined.

For example, take ''a'' = −1. The ''n''th root of −1
is −1 for every odd natural number ''n''. So if ''n'' is an odd positive integer, (−1)^{(''m''/''n'')} = −1 if ''m'' is odd, and (−1)^{(''m''/''n'')} = 1 if ''m'' is even. Thus the set of rational numbers ''q'' for which −1^''q'' = 1 is Dense in the rational numbers, as is the set of ''q'' for which −1^''q'' = −1. This means that the function (−1)^''q'' is not continuous at any rational number ''q'' where it is defined.


Imaginary powers of e


The Geometric Interpretation Of The Operations On Complex Numbers and the definition of Powers Of E is the clue to understanding ''e'' ^''i·x'' for real ''x''. Consider the Right Triangle (0, 1, 1+''i·x''/''n''). For big values of ''n'' the triangle is almost a Circular Sector with a small central angle equal to ''x''/''n'' radian. The triangles (0, (1+''i·x/n'')^''k'', (1+''i·x''/''n'')^''k''+1) are mutually Similar for all values of ''k''. So for big values of ''n'' the limiting point of (1+''ix/n'')^''n'' is the point on the Unit Circle whose angle from the positive real axis is ''x'' Radian s. The Polar Coordinates of this point are (''r'',''θ'') = (1,''x''), and the Cartesian Coordinate s are (cos(''x''), sin(''x'')). So ''e'' ^''i·x'' = cos(''x'') + ''i''·sin(''x''), and this is Euler's Formula , connecting Algebra to Trigonometry by means of Complex Number s.

The solutions to the equation e^''z'' = 1 are the integer multiples of 2·π·''i'' :
:$\backslash \{\; z\; :\; e^z=1\; \backslash \}\; =\; \backslash \{\; k\backslash cdot\; 2\backslash cdot\; \backslash pi\backslash cdot\; i\; :\; k\; \backslash in\; \backslash mathbb\{Z\}\; \backslash \}.$
More generally, if e^''b'' = ''a'', then every solution to ''e''^''z'' = ''a'' can be obtained by adding an integer multiple of 2·π·''i'' to ''b'':
:$\backslash \{\; z\; :\; e^z=a\; \backslash \}\; =\; \backslash \{\; b+k\backslash cdot\; 2\backslash cdot\backslash pi\backslash cdot\; i\; :\; k\; \backslash in\; \backslash mathbb\{Z\}\; \backslash \}$.
Thus the complex exponential function is a Periodic Function with period 2·π·''i''.


Trigonometric functions

See Also: Euler's formula


It follows from Euler's Formula that the Trigonometric Functions cosine and sine are

:

Historically, cosine and sine were defined geometrically before the invention of complex numbers. The above formula reduces the complicated formulas for Trigonometric Functions Of A Sum into the simple exponentiation formula
:$e^\{i\backslash cdot\; (x+y)\}=e^\{i\backslash cdot\; x\}\backslash cdot\; e^\{i\backslash cdot\; y\}.\backslash ,$
Using exponentiation with complex exponents one need not study trigonometry.


Complex powers of e

The power ''e'' ^{''x''+''i''·''y''} is computed ''e'' ^''x'' · ''e'' ^''i''·''y''. The real factor ''e'' ^''x'' is the Absolute Value of ''e'' ^{''x''+''i''·''y''} and the complex factor ''e'' ^''i''·''y'' identifies the Direction of ''e'' ^{''x''+''i''·''y''}.


Complex powers of positive real numbers


If ''a'' is a positive real number, and ''z'' is any complex number, the power ''a''^''z'' is defined as ''e''^{''z''·log(''a'')}, where ''x'' = log(''a'') is the unique real solution to the equation ''e''^''x'' = ''a''. So the same method working for real exponents also work for complex exponents.
For example:
:2 ^''i'' = ''e'' ^{''i''·log(2)} = cos(log(2))+''i''·sin(log(2)) = 0.54030+''i''·0.84147

:10 ^''i'' = −0.66820+''i''·0.74398
:(''e'' ^2·''π'') ^''i'' = 535.49 ^''i'' = 1


POWERS OF COMPLEX NUMBERS


Integer powers of complex numbers are defined by repeated multiplication or division as above. Complex powers of positive reals are defined via ''e''^''x'' as above. These are continuous functions. Trying to extend these functions to the general case of non-integer powers of complex numbers that are not positive reals leads to difficulties. Either we define discontinuous functions or Multivalued Function s. None of these options are entirely satisfactory.

The rational power of a complex number must be the solution to an algebraic equation. For example, ''w'' = ''z''^1/2 must be a solution to the equation ''w''² = ''z''. But if ''w'' is a solution, then so is −''w'', because (−1)² = 1 . So the algebraic equation ''w''² = ''z'' is not sufficient for defining ''z''^1/2 . Choosing one of the two solutions as the ''principal'' value of ''z''^1/2 leaves us with a function that is not continuous, and the usual rules for manipulating powers lead us astray.


The logarithm of a complex numbers


One solution, ''z''=log ''a'', to the equation e^''z'' = ''a'', is called the Principal Value of the complex logarithm. It is the unique solution whose imaginary part lies in the Interval $(-\backslash pi,\; \backslash pi]\backslash ,$.

For example, log(1) = 0, log(−1) = π''i'', log(''i'') = π''i''/2, and log(−''i'') = −π''i''/2.
The principal value of the logarithm is known as a branch of the logarithm; other branches can be specified by choosing a different range for the imaginary part of the logarithm. The boundary between branches is known as a Branch Cut . The principal value has a branch cut extending from the origin along the negative real axis, and is discontinuous at each point of the branch cut.


Complex power of a complex number


The general complex power ''a''^''b'' of a nonzero complex number ''a'' is defined as
:$a^b\; =\; e^\{\backslash log(a^b)\}\; =\; e^\{b\backslash cdot\; \backslash log\; a\}.\backslash ,$

When the exponent is a Rational Number the power ''z''=''a''^''n''/''m'' is a solution to the equation ''z''^''m'' = ''a''^''n'' .

The computation of complex powers is facilitated by converting the base ''a'' to polar form, as described in detail Below .


Complex roots of unity

See Also: Root of unity



A complex number ''a'' such that ''a''^''n'' = 1 for a positive integer ''n'' is an ''n''th root of unity. Geometrically, the ''n''th roots of unity lie on the unit circle of the complex plane at the vertices of a regular ''n''-gon with one vertex on the real number 1.

If ''z'' ^''n'' = 1 but ''z'' ^''k'' ≠ 1 for all natural numbers ''k'' such that 0 < ''k'' < ''n'', then ''z'' is called a primitive ''n''th root of unity. The negative unit −1 is the only primitive square root of unity. The Imaginary Unit ''i'' is one of the two primitive 4'th roots of unity; the other one is −''i''.

The number ''e''^{2''πi'' (1/''n'')} is the primitive ''n''th root of unity with the smallest positive Complex Argument . (It is sometimes called the principal ''n''th root of unity, although this terminology is not universal and should not be confused with the Principal Value of $\backslash sqrt${Link without Title} {1}, which is 1.This definition of a principal root of unity can be found in:

• ¹ Online resource


• ² Defined on page 351, available on Google books.


• " Principal root of unity ", MathWorld.)

The other ''n''th roots of unity are given by
:$\backslash big\; (\; e^\{2\backslash pi\; i\; (1/n)\}\; \backslash big\; )\; ^k\; =\; e^\{2\backslash pi\; i\; k/n\}$
for 2 ≤ k ≤ n.


Roots of arbitrary complex numbers

Although there are infinitely many possible values for a general complex logarithm, there are only a finite number of values for the power ''a^z'' in the important special case where ''z'' = 1/''n'' and ''n'' is a positive integer. These are the ''n''th roots of ''a''; they are solutions of the equation ''xⁿ'' = ''a''. As with real roots, a second root is also called a square root and a third root is also called a cube root.

It is conventional in mathematics to define ''a''^1/''n'' as the principal value of the root. If ''a'' is a positive real number, it is also conventional to select a positive real number as the principal value of the root ''a''^1/''n''. For general complex numbers, the ''n''th root with the smallest argument is often selected as the principal value of the ''n''th root operation, as with principal values of roots of unity.

The set of ''n''th roots of a complex number ''a'' is obtained by multiplying the principal value ''a''^1/n by each of the ''n''th roots of unity. For example, the fourth roots of 16 are 2, −2, 2''i'', and −2''i'', because the principal value of the fourth root of 16 is 2 and the fourth roots of unity are 1, −1, ''i'', and −''i''.


Computing complex powers

See Also: Polar coordinate system


It is often easier to compute complex powers by writing the number to be exponentiated in Polar Form . Every complex number ''z'' can be written in the polar form
:$z\; =\; re^\{i\; heta\}\; =\; e^\{\backslash log(r)\; +\; i\; heta\}\; \backslash ,,$
where ''r'' is a non-negative real number and θ is the (real) using Cartesian Coordinates , then (''r'',θ) is the same point in Polar Coordinates . That is, ''r'' is the "radius" ''r''²=''u''²+''v''² and θ is the "angle" θ= Atan2 (v,u). The branch cut corresponds to the notion that a polar angle θ is ambiguous, since any multiple of 2π could be added to θ without changing the location of the point. The principal value (the most common branch cut), as mentioned above, corresponds to θ chosen in the interval (−π, π].

In order to compute the complex power ''a''^''b'', write ''a'' in polar form:
:$a\; =\; r\; e^\{i\; heta\}\; \backslash ,$.
Then
:$\backslash log\; a\; =\; \backslash log\; r\; +\; i\; heta\; \backslash ,,$
and thus
:$a^b\; =\; e^\{b\; \backslash log\; a\}\; =\; e^\{b(\backslash log\; r\; +\; i\; heta)\}.\; \backslash ,$
If ''b'' is decomposed as ''c'' + ''di'', then the formula for ''a''^''b'' can be written more explicitly as
:$\backslash left(\; r^c\; e^\{-d\; heta\}\; ight)\; e^\{i\; (d\; \backslash log\; r\; +\; c\; heta)\}\; =\; \backslash left(\; r^c\; e^\{-d\; heta\}\; ight)\; \backslash left\backslash cos(d\; \backslash log\; r\; +\; c\; heta)\; +\; i\; \backslash sin(d\; \backslash log\; r\; +\; c\; heta)\; ight.$
This final formula allows complex powers to be computed easily from decompositions of the base into polar form and the exponent into Cartesian form. It is shown here both in polar form and in Cartesian form (via Euler's identity).

The following examples use the principal value (the branch cut which causes θ to be in the interval (−π, π]). To compute ''i'' ^''i'', write ''i'' in polar and Cartesian forms:
:$i\; =\; 1\; \backslash cdot\; e^\{i\; \backslash pi/2\},\backslash ,$
:$i\; =\; 0\; +\; 1i.\backslash ,$
Then the formula above, with ''r'' = 1, θ = π/2, ''c'' = 0, and ''d'' = 1, yields:
:$\backslash \; i^i\; =\; \backslash left(\; 1^0\; e^\{-\backslash pi/2\}\; ight)\; e^\{i(1\backslash cdot\; \backslash log\; 1\; +\; 0\; \backslash cdot\; \backslash pi/2)\}\; =\; e^\{-\backslash pi/2\}\; \backslash approx\; 0.2079.$

Similarly, to find (−2)^{3 + 4''i''}, compute the polar form of −2,

:$-2\; =\; 2e^\{i\; \backslash pi\}\; \backslash ,\; ,$
and use the formula above to compute
:$(-2)^\{3+4i\}\; =\; \backslash left(\; 2^3\; e^\{-4\backslash pi\}\; ight)\; e^\{i(4\backslash log(2)\; +\; 3\backslash pi)\}\; \backslash approx\; (2.602\; -\; 1.006\; i)\; \backslash cdot\; 10^\{-5\}.$

The value of a complex power depends on the branch used. For example, if the polar form ''i'' = 1''e''^''i''(5π/2) is used to compute ''i'' ^''i'', the power is found to be ''e''^−5π/2; the principal value of ''i'' ^''i'', computed above, is ''e''^−π/2.


Failure of power and logarithm identities


Identities for powers and logarithms that hold for positive real numbers may fail when the positive real numbers are replaced by arbitrary complex numbers. There is no method to define complex powers or the complex logarithm as complex-valued functions while preserving the identities these operations possess in the positive real numbers.

An example involving logarithms concerns the rule log(''a''^''b'') = ''b''·log(''a''), which holds whenever ''a'' is a positive real number and ''b'' is a real number. The following calculation shows that this identity does not hold in general for the principal value of the complex logarithm when ''a'' is not a positive real number:
:$i\backslash pi\; =\; \backslash log(-1)\; =\; \backslash log((-i)^2)$
ot = 2\log(-i) = 2(-i\pi/2) = -i\pi.
Regardless of which branch of the logarithm is used, a similar failure of the identity will always exist.

An example involving power rules concerns the identities
:
These identities are valid when ''a'' and ''b'' are positive real numbers and ''c'' is a real number. But a calculation using principal values shows that
:$1\; =\; (-1\backslash cdot\; -1)^\{1/2\}$
ot = (-1)^{1/2}(-1)^{1/2} = -1,
and
:
ot = rac{1^{1/2}}{(-1)^{1/2}} = rac{1}{i} = -i.

These examples illustrate that complex powers and logarithms do not behave the same way as their real counterparts, and so caution is required when working with the complex versions of these operations.


ZERO TO THE ZERO POWER


The evaluation of 0⁰ presents a problem, because different mathematical reasoning leads to different results. The best choice for its value depends on the context. According to Benson (1999), "The choice whether to define 0⁰ is based on convenience, not on correctness."Benson, Donald C. ''The Moment of Proof : Mathematical Epiphanies.'' New York Oxford University Press (UK), 1999. ISBN 9780195117219 There are two principal treatments in practice, one from discrete mathematics and the other from analysis.

In many settings, especially in foundations and combinatorics, 0⁰ is defined to be 1. This definition arises in foundational treatments of the natural numbers as Finite Cardinals , and is useful for shortening combinatorial identities and removing special cases from theorems, as illustrated below. In many other settings, 0⁰ is left undefined. In Calculus , 0⁰ is an Indeterminate Form , which must be analyzed rather than evaluated. In general, Mathematical Analysis treats 0⁰ as Undefined Examples of this include Edwards and Penny (1994). ''Calculus'', 4th ed,, Prentice-Hall, p. 466, and Keedy, Bittinger, and Smith (1982). ''Algebra Two.'' Addison-Wesley, p. 32. in order that the exponential function be Continuous .

Justifications for defining 0⁰ = 1 include:

• $x^x\; o\; 1\; \backslash ,$ as $x\; o\; 0\backslash ,$


• When 0⁰ is regarded as an Empty Product of zeros, its value is 1.


• The Combinatorial Interpretation of 0⁰ is the number of empty tuples of elements from the empty set. There is exactly one empty Tuple .


• Equivalently, the Set-theoretic Interpretation of 0⁰ is the number of functions from the empty set to the empty set. There is exactly one such function, the Empty Function .


• A Power Series identity with nonzero constant term, such as , is not valid for ''x'' = 0 unless 0⁰, which appears in the numerator of the first term of the series, is 1. Thus defining 0⁰ to be 1 allows this identity to be used instead of the longer identity .


• The Binomial Theorem $extstyle(1+x)^n\; =\; \backslash sum\_\{k\; =\; 0\}^n\; \backslash binom\{n\}\{k\}\; x^k$ is not valid for ''x'' = 0, unless 0⁰ = 1."Some textbooks leave the quantity 0⁰ undefined, because the functions ''x''⁰ and 0^''x'' have different limiting values when ''x'' decreases to 0. But this is a mistake. We must define ''x''⁰ = 1, for all ''x'', if the binomial theorem is to be valid when ''x'' = 0, ''y'' = 0, and/or ''x'' = −''y''. The binomial theorem is too important to be arbitrarily restricted! By contrast, the function 0^''x'' is quite unimportant".³ By defining 0⁰ to be 1, a special case of the theorem can be eliminated.

In contexts where the exponent may vary continuously, it is generally best to treat 0⁰ as an ill-defined quantity. Justifications for treating it as undefined include:

• The value 0⁰ often arises as the formal limit of exponentiated functions, ''f''(''x'')^''g''(''x''), when ''f''(''x'') and ''g''(''x'') approach 0 as ''x'' approaches ''a'' (a constant or infinity). There, 0⁰ suggests ''f''(''x'') ^{lim ''g''(''x'')}, which is a well defined quantity and is the correct value of lim ''f''(''x'')^''g''(''x'') when ''f'' and ''g'' approach nonzero constants, but is not well defined when ''f'' and ''g'' approach 0. The same reasoning applies to certain powers involving Infinity , $\backslash infty^0$ and $1^\backslash infty$.

:A more abstract way of saying this is the following: The real function ''x''^''y'' of the two nonnegative real variables ''x'' and ''y'' is not Continuous at the point (''x'', ''y'') = (0, 0), and so 0⁰ is not determined by continuity.⁴ That is, the function ''x''^''y'' has no continuous extension from the open first quadrant to include the point (0,0).Along the ''x''-axis the limit is 1, along the ''y''-axis the limit is 0, and any intermediate limit ''a'' can be obtained using the curve ''y'' = log(''a'')/log(''x''). However, if ''y'' is an Analytic Function of ''x'', or if there exists a positive constant, ''a'', such that ''y'' < ''ax'', then the limit is 1. A ''discontinuous'' extension would cause the function to lose a number of desirable properties. For example, it is ordinarily taken as a rule in calculus that $\backslash lim\_\{x\; o\; a\}\; f(x)^\{g(x)\}\; =$

(\lim_{x o a} f(x))^{\lim_{x o a} g(x)} whenever both sides of the equation are defined; this rule would necessarily fail if 0⁰ were defined.

• The function ''z''^''z'', viewed as a function of a Complex Number variable ''z'' and defined as ''e''^{''z'' log ''z''} is undefined at ''z'' = 0 because log ''z'' is undefined at ''z'' = 0. Moreover, because ''z''^''z'' has a logarithmic Branch Point at ''z'' = 0, it is not common to extend the domain of ''z''^''z'' to the origin in this context."... Let's start at ''x''=0. Here ''x''^''x'' is undefined." Mark D. Meyerson "The ''X''^''x'' Spindle." ''Mathematics Magazine'', v. 69 n. 3, Jun 1996, pp. 198-206.

Treatment in programming languages

The evaluation of 0⁰ is possible in several computer programming languages. Many languages, including Java , Python , Ruby , Haskell , ML , Scheme , MATLAB , Microsoft Windows' Calculator, and others (especially when using IEEE Floating-point Arithmetic , but also for integer arithmetic), evaluate 0⁰ to be 1.For example, see ⁵ Mathematica simplifies ''a''⁰ to 1, even if no constraints are placed on ''a'', but does not simplify 0^''a'', and takes 0⁰ to be an indeterminate form. Maple simplifies ''a''⁰ to 1 and 0^''a'' to 0, even if no constraints are placed on ''a'', and evaluates 0⁰ to 1. Google search when used for its calculator function evaluates ''0''⁰ to 1 http://www.google.co.uk/search?q=0%5E0


POWERS WITH INFINITY

The expressions $\backslash infty^0$ and $1^\backslash infty$ arise in analysis for the same reason as 0⁰, and they are undefined for the same reason. That is, it is true that lim ''f''(''x'')]^{lim ''g''(''x'')} = lim ''f''(''x'')^''g''(''x'') when ''f'' and ''g'' approach nonzero finite constants, but not when they approach 0 or infinity; then, the limit of the power can be anything, not predictable from the limits of ''f'' and ''g''.

There is one exception. If ''f'' and ''g'' both approach infinity as ''x'' approaches ''a'', then lim ''f''(''x'')^''g''(''x'') does equal infinity. Thus, it makes sense to say the expression $\backslash infty^\backslash infty$ is well defined but, by Cantor's Theorem , does not equal the ''same'' $\backslash infty$; instead we have and hence . Thus exponentiation defines an infinite hierarchy of infinities, usually expressed as Beth Number s.


EFFICIENTLY COMPUTING A POWER

The simplest method of computing ''a''^''n'' requires ''n''−1 multiplication operations, but it can be computed more efficiently as illustrated by the following example. To compute 2¹⁰⁰, note that 100 = 96 + 4 and

• 32. Compute the following in order:

:2² = 4

:(2²)² = 2⁴ = 16
:(2⁴)² = 2⁸ = 256
:(2⁸)² = 2¹⁶ = 65,536
:(2¹⁶)² = 2³² = 4,294,967,296
: 2³² 2³² 2³²2⁴ = 2¹⁰⁰
This series of steps only requires 8 multiplication operations instead of 99.

In general, the number of multiplication operations required to compute
''a''^''n'' can be reduced to Θ (log ''n'') by using Exponentiation By Squaring or (more generally) Addition-chain Exponentiation . Finding the ''minimal'' sequence of multiplications (the minimal-length addition chain for the exponent) for ''a''^''n'' is a difficult problem for which no efficient algorithms are currently known, but many reasonably efficient heuristic algorithms are available.


EXPONENTIAL NOTATION FOR FUNCTION NAMES

Placing an integer superscript after the name or symbol of a function, as if the function were being raised to a power, commonly refers to repeated Function Composition rather than repeated multiplication. Thus ''f''³(''x'') may mean ''f''(''f''(''f''(''x'')));
in particular, ''f'' ^-1(''x'') usually denotes the Inverse Function of ''f''.

However, for historical reasons, a special syntax applies to the , also called arcsin ''x''. There is no need for a shorthand for the reciprocals of trigonometric functions since each has its own name and abbreviation, for example
1 / sin(''x'') = (sin ''x'')⁻¹ is csc ''x''. A similar convention applies to logarithms, where log²(''x'') = (log (''x''))² and there is no common abbreviation for log(log(''x'')).


GENERALIZATIONS OF EXPONENTIATION


Exponentiation in abstract algebra

Exponentiation for integer exponents can be defined for quite general structures in Abstract Algebra .

Let ''X'' be a Set with a Power-associative Binary Operation , which we will write multiplicatively. In this very general situation, we can define ''x''^''n'' for any element ''x'' of ''X'' and any nonzero Natural Number ''n'', by simply multiplying ''x'' by itself ''n'' times; by definition, Power Associativity means that it doesn't matter in which order we perform the multiplications.

Now additionally suppose that the operation has an Identity Element 1.
Then we can define ''x''⁰ to be equal to 1 for any ''x''.
Now ''x''^''n'' is defined for any natural number ''n'', including 0.

Finally, suppose that the operation has Inverses , and that the multiplication is associative (so that the Magma is a Group ). Then we can define ''x''^−''n'' to be the inverse of ''x''^''n'' when ''n'' is a natural number. Now ''x''^''n'' is defined for any integer ''n'' and any ''x'' in the group.

Exponentiation in this purely algebraic sense satisfies the following laws (whenever both sides are defined):

• $\backslash \; x^\{m+n\}=x^mx^n$


• $\backslash \; x^\{m-n\}=x^m/x^n$


• $\backslash \; x^\{-n\}=1/x^n$


• $\backslash \; x^0=1$


• $\backslash \; x^1=x$


• $\backslash \; x^\{-1\}=1/x$


• $\backslash \; (x^m)^n=x^\{mn\}$

Here, we use a Division slash ("/") to indicate multiplying by an inverse, in order to reserve the symbol ''x''⁻¹ for raising ''x'' to the power −1, rather than the inverse of ''x''. However, as one of the laws above states, ''x''⁻¹ is always equal to the inverse of ''x'', so the notation doesn't matter in the end.

If in addition the multiplication operation is Commutative (so that the set ''X'' is an Abelian Group ), then we have some additional laws:

• (''xy'')^''n'' = ''x''^''n''''y''^''n''


• (''x''/''y'')^''n'' = ''x''^''n''/''y''^''n''

If we take this whole theory of exponentiation in an algebraic context but write the binary operation additively, then "exponentiation is repeated multiplication" can be reinterpreted as " Multiplication is repeated Addition ". Thus, each of the laws of exponentiation above has an Analogue among laws of multiplication.

• ''n'' is ''x'' --- ··· --- ''x'', while ''x''^#''n'' is ''x'' # ··· # ''x'', whatever the operations --- and # might be.

Superscript notation is also used, especially in Group Theory , to indicate Conjugation . That is, ''g''^''h'' = ''h''⁻¹''gh'', where ''g'' and ''h'' are elements of some Group . Although conjugation obeys some of the same laws as exponentiation, it is not an example of repeated multiplication in any sense. A Quandle is an Algebraic Structure in which these laws of conjugation play a central role.


Exponentiation over sets


If ''n'' is a natural number and ''A'' is an arbitrary set, the expression ''A''^''n'' is often used to denote the set of ordered ''n''-tuples of elements of ''A''. This is equivalent to letting ''A''^''n'' denote the set of functions from the set {0, 1, 2, ..., ''n''−1} to the set ''A''; the ''n''-tuple (''a''₀, ''a''₁, ''a''₂, ..., a_''n''−1) represents the function that sends ''i'' to ''a''_''i''.

For an infinite Cardinal Number κ and a set ''A'', the notation ''A''^κ is also used to denote the set of all functions from a set of size κ to ''A''. This is sometimes written ^κ''A'' to distinguish it from cardinal exponentiation, defined below.

This generalized exponential can also be defined for operations on sets or for sets with extra Structure . For example, in Linear Algebra , it makes sense to index Direct Sum s of Vector Space s over arbitrary index sets.
That is, we can speak of
: $\backslash bigoplus\_\{i\; \backslash in\; \backslash mathbb\{N\}\}\; V\_\{i\},$
where each ''V''_''i'' is a vector space.
Then if ''V''_''i'' = ''V'' for each ''i'', the resulting direct sum can be written in exponential notation as ''V''^(+)''N'', or simply ''V''^''N'' with the understanding that the direct sum is the default.
We can again replace the set ''N'' with a cardinal number ''n'' to get ''V''^''n'', although without choosing a specific standard set with cardinality ''n'', this is defined only Up To Isomorphism .
Taking ''V'' to be the Field R of Real Number s (thought of as a vector space over itself) and ''n'' to be some Natural Number , we get the vector space that is most commonly studied in linear algebra, the Euclidean Space R^''n''.

If the base of the exponentiation operation is a set, the exponentiation operation is the Cartesian Product unless otherwise stated. Since multiple Cartesian products produce an ''n''- Tuple , which can be represented by a function on a set of appropriate cardinality, ''S''^''N'' becomes simply the set of all Function s from ''N'' to ''S'' in this case:
: $S^N\; \backslash equiv\; \backslash \{\; f:\; N\; o\; S\; \backslash \}\backslash ,$
  When ''N'' 2={0,1}, we have 2^''X'' = 2^''X'', where 2^''X'', usually denoted by '''P'''''X'', is the Power Set of ''X'' each Subset ''Y'' of ''X'' corresponds uniquely to a function on ''X'' taking the value 1 for ''x''&nbsp&isin&nbsp''Y'' and 0 for ''x''&nbsp&notin&nbsp''Y''