Negative integer exponents

a⁻¹ = 1/a

Thus:

a⁻ⁿ = (aⁿ)⁻¹ = 1/aⁿ

A negative integer exponent can also be seen as repeated division by the base. Thus 3⁻⁵ = 1 ÷ 3 ÷ 3 ÷ 3 ÷ 3 ÷ 3 = 1/243 = 1/3⁵.

Identities and properties

The most important identity satisfied by integer exponentiation is:

$a^\{m\; +\; n\}\; =\; a^m\; \backslash cdot\; a^n$

$a^\{m\; -\; n\}\; =\; \backslash begin\{matrix\}\backslash frac\{a^m\}\{a^n\}\backslash end\{matrix\}$

$(a^m)^n\; =\; a^\{mn\}\; \backslash !\backslash ,$

Whereas addition or multiplication are commutative (for example, 2+3 = 5 = 3+2 and 2×3 = 6 = 3×2), exponentiation is not commutative: 2³ = 8 while 3² = 9. Similarly, whereas addition or 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, while 2 to the 3⁴ power is 2⁸¹ or 2,417,851,639,229,258,349,412,352.

Powers of ten

Powers of 10 are easy to compute in the base ten (decimal) number system: for example 10⁶ = 1 million, which is 1 followed by 6 zeros. Exponentiation with base 10 is often used in the physical sciences to describe large or small numbers in scientific notation; for example, 299792458 (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 if this is useful. SI prefixes are also used to describe small or large quantities, and these are also based on powers of 10; for example, the prefix kilo means 10³ = 1000, so a kilometre is 1000 metres.

Powers of two

The positive powers of 2 are important in computer science because there are 2ⁿ possible values for a n bit variable. See Binary numeral system.

SI decimal prefixes were confusingly reinterpreted to refer to binary: 1 kilobyte = 2¹⁰ = 1024 bytes. In 1998 the International Electrotechnical Commission approved a set of binary prefixes. The prefix for multiples of 1024 is kibi-, so 1024 bytes is 1 kibibyte. Other prefixes are mebi-, gibi-, and tebi-.

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

If the exponent is zero, the power is commonly considered undefined.^* At times, however, it is useful to define 0⁰ = 1. (Note that y^x is discontinuous for x=y=0, and so $\backslash lim\_\{x,y\backslash rarr\; 0\}y^x$ is not necessarily equal to 0⁰. See also Empty product.)

Powers of minus one

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

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

Powers of i

i⁴ⁿ⁺¹ = i

i⁴ⁿ⁺² = −1

i⁴ⁿ⁺³ = −i

i⁴ⁿ⁺⁴ = 1

Powers of e

$\backslash \; e=\backslash lim\_\{n\; \backslash rightarrow\; +\backslash infty\}\; \backslash left(1+\backslash frac\{1\}\{n\}\; \backslash right)\; ^n\; =\backslash lim\_\{n\; \backslash rightarrow\; -\backslash infty\}\; \backslash left(1+\backslash frac\{1\}\{n\}\; \backslash right)\; ^n$.

$\backslash \; e\backslash approx\; 2.71828$.

A non-zero integer power of e is

$e^x$

The right hand side generalizes the meaning of e^x so that x does not have to be a non-zero integer but can be zero, a fraction, a real number, a complex number, or a square matrix.

(See Characterizations of the exponential function for equivalent alternate definitions.)

Real powers of positive real numbers

• Defining fractional exponents in terms of n^th roots. This method is perhaps the way most widely taught in schools.
• Defining the natural logarithm as the area under the curve 1/x.

The identities and properties shown above are true for non-integer exponents as well.

Fractional exponent

If $\backslash \; a$ is a positive real number, and n is a positive integer, then the positive real solution to the equation

$\backslash \; x^n\; =\; a$

$x=a^\{\backslash frac\{1\}\{n\}\}$

For example: 8^1/3 = 2.

Exponentiation with a rational exponent $m/n$ can now be defined as

$a^\{\backslash frac\{m\}\{n\}\}\; =\; \backslash left(a^\{\backslash frac\{1\}\{n\}\}\backslash right)^m$

Since any real number can be approximated by rational numbers, exponentiation to an arbitrary real exponent can be defined by continuity. For example, if

$k\; \backslash approx\; 1.732$

$5^k\; \backslash approx\; 5^\{1.732\}$

Logarithm method

If a and b are positive real numbers, then the real solution x to the equation

b^x = a

x = log_b(a)

Define the natural logarithm, ln, of a positive real number, a, as the area under the curve 1/x between from x = 1 to x = a. (The area is negative if a < 1). In terms of integral calculus:

$\backslash ln(a)\; =\; \backslash log\_e(a)\; =\; \backslash int\_1^a\; \backslash frac\; \{dx\}\{x\}$

The exponential function e^x is the inverse function to the natural logarithm.

a = e^ln(a)

b^x = e^{x ln(b)}

Complex powers of complex numbers

Summary

z⁰ = 1

zⁿ⁺¹ = z·zⁿ

z⁻ⁿ = 1/zⁿ (for z ≠ 0).

Complex powers of e was defined above.

$e^z=\backslash lim\_\{n\backslash rarr\backslash infty\}\backslash left(1+\backslash frac\{z\}\{n\}\backslash right)^n$

Complex powers of a complex number:

a^z = e^bz

a = e^b

Trigonometry

$\backslash \; e^\{ix\}=\backslash cos(x)\; +\; i\; \backslash sin(x)$

$\backslash \; e^\{-ix\}=\backslash cos(x)\; -\; i\; \backslash sin(x)$

$\backslash \; \backslash cos(x)\; =\; (e^\{ix\}\; +\; e^\{-ix\})\; /\; \{2\}$

$\backslash \; \backslash sin(x)\; =\; (e^\{ix\}\; -\; e^\{-ix\})\; /\; \{2i\}$

Primitive and principal logarithms of unity

e^2πi = e⁰ = 1.

Root of unity

Multivalued logarithm

e^x+2πi·n = e^x·e^2πi·n = e^x·(e^2πi)ⁿ = e^x·1ⁿ = e^x·1 = e^x

So the logarithm is a Multivalued function.

Singlevalued logarithm

Multivalued power

Singlevalued power

Polar form

$a+ib\; =\; r\; e^\{i\backslash varphi\}\; =\; r\; \backslash left\backslash cos\backslash varphi\; +\; i\; \backslash sin\backslash varphi\; \backslash right$

for a positive real magnitude $r$ and a real angle $\backslash varphi$, where for the right-most equation we have used Euler's formula for $e^\{i\backslash varphi\}$. Then, exponentiation can be written as:

$(a+ib)^x\; =\; \backslash left(\; r\; e^\{i\backslash varphi\}\; \backslash right)^x\; =\; r^x\; e^\{i\; \backslash varphi\; x\}.$

For real $x$, $r^x$ is handled as above. For complex $x$, we use Euler's formula a second time as explained below.

As for real numbers, above, any non-integer exponent $x$ implies that the answer is not uniquely determined. In particular, we could change $\backslash varphi$ to $\backslash varphi\; +\; 2\backslash pi\; n$ (see Pi) for any integer $n$ without changing the formula for $a+ib$, since $e^\{i\; 2\backslash pi\; n\}=1$ by Euler's formula. Different values of $n$ may change the exponential, however, since $e^\{i\; 2\backslash pi\; n\; x\}\backslash neq\; 1$ in general. For a rational real x, the number of possible values is given by the lowest common denominator of x (see Root of unity), while for other real or complex x there are infinitely many possible values.

By convention, this multi-valuedness is resolved by defining $(a+ib)^x$ as the principal value, as for real exponentials above, unless otherwise noted. This means that the angle $\backslash varphi$ is conventionally chosen to lie in the interval $(-\backslash pi,\backslash pi]$.

In the above, we didn't explain how to handle one important case: how do we compute the exponential when $x=c+id$ is complex? In particular, we now have to take the complex exponential $r^x\; =\; r^c\; r^\{id\}\; \backslash !$ of a positive real number $r$. $r^c\; \backslash !$ is purely real and is the same as above, so we only need to understand $r^\{id\}\; \backslash !$.

Here, we can once again exploit Euler's formula, since it tells us how to take imaginary powers of one real number e: $e^\{id\}\; =\; \backslash cos\; d\; +\; i\backslash sin\; d$. Therefore, we just need to rewrite $r^\{id\}$ in terms of a power of e:

$r^\{id\}\; =\; \backslash left(r)^d\; \backslash right^i\; =\; \backslash left\backslash left(\; e^\{\backslash ln\; r\}\; \backslash right)^d\; \backslash right^i\; =\; e^\{i\; d\; \backslash ln\; r\}\; =\; \backslash cos(d\; \backslash ln\; r)\; +\; i\backslash sin(d\; \backslash ln\; r).$

Here, as we did above for real exponents, we used the natural logarithm function ln to write $r\; =\; e^\{\backslash ln\; r\}\; \backslash !$.

So, we can finally write:

$(a+ib)^\{c+id\}\; =\; \backslash left(\; r\; e^\{i\backslash varphi\}\; \backslash right)^\{c+id\}\; =\; \backslash leftr^c\; e^\{-\backslash varphi\; d\}\; \backslash righte^\{i(\backslash varphi\; c\; +\; d\; \backslash ln\; r)\}$

where we have written the final expression in polar form as a real magnitude multiplied by a complex phase, and have used the fact that $i\backslash cdot\; i\; =\; -1$.

Examples

$i^i\; =\; (e^\{i\backslash pi/2\})^i\; =\; e^\{-\backslash pi/2\}\; \backslash approx\; 0.20788\backslash ldots$

This is the principal value of $i^i$. One could also write $i\; =\; e^\{i\backslash pi/2\; +\; 2\backslash pi\; i\backslash cdot\; n\}$ for any integer n, resulting in an infinite set of possible definitions

$i^i\; =\; (e^\{i\backslash pi/2\; +\; 2\backslash pi\; i\backslash cdot\; n\})^i\; =\; e^\{-\backslash pi/2\; -\; 2\backslash pi\backslash cdot\; n\}$

However, according to standard conventions the expression $i^i$ denotes the principal value ($n=0$) unless otherwise specified.

In the same way, one can define exponentiation of negative real numbers, since any negative real number $-r$ can be written:

$-r\; =\; r\; e^\{i\backslash pi\}\; \backslash !$

and thus the principal value of the exponent is $(-r)^x\; =\; r^x\; e^\{i\backslash pi\; x\}\; \backslash !$ (with the exponential $r^x$ of a positive number $r$ defined above).

Solving polynomial equations

It was once conjectured that the roots of any polynomial could be expressed in terms of exponentiation with fractional exponents. (See Quadratic equation).

That this is not true in general is the assertion of the Abel-Ruffini theorem.

For example, the solutions of the equation x⁵ = x+1 cannot be expressed in terms of fractional exponents.

For solving any equation of the n^th degree, see the Durand-Kerner method.

Advanced topics

Efficient computing

a¹⁵ = a·a¹⁴ = a·(a⁷)² = a·(a·a⁶)² = a·(a·(a³)²)² = a·(a·(a·a²)²)²

a¹⁵ = a³·((a³)²)²

Exponents on function names

When the name or symbol of a function is given an integer superscript, as if being raised to a power, this commonly refers to repeated function composition rather than repeated multiplication. Thus f³(x) may mean f(f(f(x))); in particular, f ^-1(x) usually denotes f's inverse function.

A special syntax applies to the trigonometric functions: a positive exponent applied to the function's abbreviation means that the result is raised to that power, while an exponent of -1 indicates the inverse function. That is, sin²x is just a shorthand way to write (sin x)² without using parentheses, whereas sin^-1x refers to the inverse function of the sine, also called arcsin x. There is no need for a shorthand of this kind for reciprocal trigonometric functions since they each have their own name and abbreviation already: (sin x)^-1 is normally just written as csc x.

Exponentiation in abstract algebra

Exponentiation can also be understood purely in terms of abstract algebra, if we limit the exponents to integers.

Specifically, suppose that X is a set with a power-associative binary operation, which we will write multiplicatively. In this very general situation, we can define xⁿ 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 x⁰ to be equal to 1 for any x. Now xⁿ 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 xⁿ when n is a natural number. Now xⁿ is defined for any integer n.

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

• $\backslash \; x^\{m+n\}=x^mx^n$
• $\backslash \; x^\{m-n\}=x^m/x^n$
• $\backslash \; x^\{-n\}=1/x^n$
• $\backslash \; x^0=1$
• $\backslash \; x^1=x$
• $\backslash \; x^\{-1\}=1/x$
• $\backslash \; (x^m)^n=x^\{mn\}$

If in addition the multiplication operation is commutative (so that the magma is an abelian group), then we have some additional laws:

• (xy)ⁿ = xⁿyⁿ
• (x/y)ⁿ = xⁿ/yⁿ

It is necessary to let 0⁰ be 1, just like every other case of x⁰. For example, expanding (0 + x)ⁿ by the binomial theorem, it is necessary to take 0⁰ = 1.

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.

When one has several operations around, any of which might be repeated using exponentiation, it is common to indicate which operation is being repeated by placing its symbol in the superscript. Thus, x^*n is x * ··· * x, while x^#n is x # ··· # x, whatever the operations * and # might be.

Exponential notation is also used, especially in group theory, to indicate conjugation. That is, g^h = h^-1gh, 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

The above algebraic treatment of exponentiation builds a finitary operation out of a binary operation. In more general contexts, one may be able to define an infinitary operation directly on an indexed set.

For example, in the arithmetic of cardinal numbers, it makes sense to consider the product

$\backslash prod\_\{i\; \backslash in\; N\}\; k\_\{i\}$

This can be done even for operations on sets or sets with extra structure. For example, in linear algebra, it makes sense to index direct sums of vector spaces over arbitrary index sets. That is, we can speak of

$\backslash bigoplus\_\{i\; \backslash in\; N\}\; V\_\{i\},$

If the base of the exponentiation operation is a set, then by default we assume the operation to be the Cartesian product. In that case, S^N becomes simply the set of all functions from N to S. This fits in with the exponentiation of cardinal numbers once gain, in the sense that |S^N| = |S|^|N|, where |X| is the cardinality of X. When N=2={0,1}, we have |2^X| = 2^|X|, where 2^X, usually denoted by PX, is the power set of X. (This is where the term "power set" comes from.)

Note that exponentiation of cardinal numbers doesn't match up with exponentiation of ordinal numbers, which is defined by a limit process. In the ordinal numbers, a^b is the smallest ordinal number greater than a^c for c < b when b is a limit ordinal, and of course a^b+1 := a^ba.

In category theory, we learn to raise any object in a wide variety of categories to the power of a set, or even to raise an object to the power of an object, using the exponential.

Exponentiation in programming languages

• x ↑ y: Algol programming language
• x ^ y: BASIC, Matlab, J programming language, Microsoft Excel and many others
• x ** y: Fortran, Perl, Python programming language, Ruby programming language, ADA programming language
• x * y: APL programming language
• Power(x, y): Excel, Pascal programming language
• pow(x, y): C programming language, C++, PHP
• Math.pow(x, y): Java programming language, JavaScript, Modula-3
• Math.Pow(x, y): C#

In C, C++, C#, Java and JavaScript, the symbol ^ represents bitwise XOR. In Pascal, it represents indirection.

Table of powers

Table of kⁿ, with k on the left and n at the top.

Generalization

See also

• List of exponential topics
• Exponential growth
• Exponential decay
• Exponentiating by squaring
• Logarithm
• Modular exponentiation
• Addition chain exponentiation using an addition chain

External links

• sci.math FAQ: What is 0⁰?
• Laws of Exponents with derivation and examples
• 1058 Powers of Two

Exponentials | Binary operations

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