Exponentiation

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Exponentiation is a mathematical operation, written aⁿ, involving two numbers, the base a and the exponent n. When n is a whole number, exponentiation is repeated multiplication:

$a^n = \underbrace{a \times \cdots \times a}_n$

just as multiplication by a whole number is repeated addition:

$a \times n = \underbrace{a + \cdots + a}_n$

Exponentiation is also known as raising the number a to the power n, or a to the n^th power, and can also be defined for exponents that are not whole numbers, as explained below.

Exponentiation with various bases: red is to base e, green is to base 10, and purple is to base 1.7. Each tick on the axes is one unit. Note how all exponentials pass through the point (0, 1).

The exponent is usually shown as a superscript to the right of the base. Exponentiation is a basic mathematical tool that is used pervasively in many other fields as well, including economics, biology, chemistry, physics, and psychology, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and allometric functions.

Exponentiation with integer exponents

The exponentiation operation with integer exponents only requires basic algebra.

Positive integer exponents

The simplest case involves a positive integer exponent. The exponent then says how many times the base is to be multiplied. For example, 3⁵ = 3 × 3 × 3 × 3 × 3 = 243. Here, 3 is the base, 5 is the exponent, and 243 is 3 raised to the fifth power or 3 raised to the power 5. (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".) Notice that the base 3 appears 5 times in the repeated multiplication, because the exponent is 5.

Traditionally a² = a×a is called the square and a³ = a×a×a is called the cube. 3² is pronounced "three squared," and 3³ is "three cubed."

Formally, powers with positive integer exponents can be defined by the initial condition a⁰ = 1 and the recurrence relation aⁿ⁺¹ = a·aⁿ .

Exponents one and zero

The meaning of 3⁵ may also be viewed as 1 × 3 × 3 × 3 × 3 × 3: the starting value 1 (the identity element of multiplication) is multiplied by the base as many times as indicated by the exponent. With this definition in mind, it is easy to see how to generalize exponentiation to exponents one and zero:

• Any number to the power 1 is itself.

$\,\!a^1=a$

• Any number to the power 0 is 1.

$\,\!a^0=a^1\cdot a^{-1}=a\cdot\frac{1}{a}=1$

(Some authors consider 0⁰ undefined. See Empty product.) Example: $\,\!a^0=a^{2-2}=\frac{a^2}{a^2}=1$, (valid when a ≠ 0).

Negative integer exponents

Raising a nonzero number to the -1 power produces its reciprocal.

a⁻¹ = 1/a

Thus:

a⁻ⁿ = (aⁿ)⁻¹ = 1/aⁿ

Raising 0 to a negative power would imply division by 0, and so is undefined.

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:

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

It has the following consequences:

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

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

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 trivial 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 (or sometimes as 299E+6, especially in computer software) 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.

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: 0ⁿ = 0, where n > 0.

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 of zero should be defined to be one: 0⁰ = 1.

(Some authors, [1], consider 0⁰ undefined. See however Empty product.)

Powers of minus one

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

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

The powers of i are useful for expressing sequences of period 4.

i⁴ⁿ⁺¹ = i

i⁴ⁿ⁺² = −1

i⁴ⁿ⁺³ = −i

i⁴ⁿ⁺⁴ = 1

Powers of e

The number e is the limit of a sequence of integer powers

$\ e=\lim_{n \rightarrow +\infty} \left(1+\frac{1}{n} \right) ^n =\lim_{n \rightarrow -\infty} \left(1+\frac{1}{n} \right) ^n.$

Approximately

$\ e\approx 2.71828.$

A non-zero integer power of e is

$e^x = \left( \lim_{m \rightarrow \pm\infty} \left(1+\frac{1}{m} \right) ^m\right) ^x = \lim_{m \rightarrow \pm\infty} \left(\left(1+\frac{1}{m} \right) ^m\right) ^x = \lim_{m \rightarrow \pm\infty} \left(1+\frac{1}{m} \right) ^{mx} = \lim_{mx \rightarrow \pm\infty} \left(1+\frac{x}{mx} \right) ^{mx} = \lim_{n \rightarrow \pm\infty} \left(1+\frac{x}{n} \right) ^n$ .

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

Raising a positive real number to a power that is not an integer can also be explained in other ways:

• 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

From top to bottom: x^1/8, x^1/4, x^1/2, x¹, x², x⁴, x⁸.

For a given exponent, the inverse of exponentiation is extracting a root.

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

$\ x^n = a$

is called the n^th root of $\ a$

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

For example: 8^1/3 = 2.

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

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

For example: 8^2/3 = 4.

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

$k \approx 1.732$

we can assume

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

Logarithm method

For a given base, the inverse of exponentiation is taking a logarithm.

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

b^x = a

is called the logarithm of a, base b.

x = log_b(a)

So, exponentiation is sometimes called the antilogarithm.

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:

$\ln(a) = \log_e(a) = \int_1^a \frac {dx}{x}$

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

a = e^ln(a)

Exponentiation in any positive real base, b, can be expressed as:

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

Complex powers of complex numbers

Summary

Integer powers of complex numbers was defined recursively above:

z⁰ = 1

zⁿ⁺¹ = z·zⁿ

z⁻ⁿ = 1/zⁿ (for z ≠ 0).

Complex powers of e was defined above.

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

Complex powers of a complex number:

a^z = e^bz

if

a = e^b

Trigonometry

From Euler's formula, the purely imaginary powers of e define the real trigonometric functions cosine and sine:

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

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

such that

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

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

Primitive and principal logarithms of unity

There exists a positive real number, π, such that any solution to the equation: e^z = 1 is of the form z = 2πi·n where n is some integer. (These logarithms of unity constitute an additive group because unity constitutes a multiplicative group.) The number 2πi = 2πi·1 is a primitive logarithm of unity, (a generator of the group), while the number 0 = 2πi·0 is the principal logarithm of unity.

e^2πi = e⁰ = 1.

Root of unity

e^2πi(1/n) is a primitive n-th root of unity, while e^2πi(0/n) is the principal n-th root of unity.

Multivalued logarithm

The equation, e^x=a, where a is a nonzero complex number, has an infinity of solutions. Let x be any of them, then any of them has the form x+2πi·n where n is some integer.

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

If a is a positive real number, then one of the solutions to the equation, e^x=a, is a real number. It is natural to select this solution as the principal value of the logarithm. In the general case the principal value of the logarithm is more arbitrarily defined as the value having imaginary part in the interval (−π,+π]. The principal value has the advantage of being singlevalued, but the price to be paid is that it ceases to be a continuous function.

Multivalued power

If e^b = a, then e^(b+2πi·n)x are the values of a^x. For example, 4^1/2 = {+2,−2}. (see square root).

Singlevalued power

If e^b = a, and b is the principal value, then e^bx is the principal value of a^x. For example, the principal value of 4^1/2 is +2.

Polar form

The typical approach is to write the complex number in polar form: any complex number $a+ib$ can be written as:

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

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

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

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 $\varphi$ to $\varphi + 2\pi n$ (see Pi) for any integer $n$ without changing the formula for $a+ib$, since $e^{i 2\pi n}=1$ by Euler's formula. Different values of $n$ may change the exponential, however, since $e^{i 2\pi n x}\neq 1$ 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 $\varphi$ is conventionally chosen to lie in the interval $(-\pi, \pi]$.

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 $r^x = r^c r^{id} \!$ of a positive real number $r$. $r^c \!$ is purely real and is the same as above, so we only need to understand $r^{id} \!$.

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

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

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

So, we can finally write:

$(a+ib)^{c+id} = \left( r e^{i\varphi} \right)^{c+id} = \left[ r^c e^{-\varphi d} \right] e^{i(\varphi c + d \ln r)}$

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 $i\cdot i=-1$.

Examples

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

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

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

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:

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

and thus the principal value of the exponent is $(-r)^{x}=r^{x}e^{i\pi x}\!$ (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

Efficiently computing exponents

It may seem that computing aⁿ requires n−1 multiplications, but this can be reduced using exponentiation by squaring or addition-chain exponentiation, both of which are types of dynamic programming.

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 for the reciprocals of trigonometric functions since each has its own name and abbreviation, for example (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):

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

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 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

$\prod_{i \in N} k_{i}$

for any indexed family of cardinal numbers, (k_i)_{i in N}. By taking k_i = k for every i, this can be interpreted as a repeated product, and the result is k^N. In fact, this result depends only on the cardinality of N, so we can define exponentiation of cardinal numbers so that kⁿ is k^N for any set N whose cardinality is n.

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

$\bigoplus_{i \in N} V_{i},$

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 Vⁿ, 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 numbers (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 Rⁿ.

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 again, 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

The mathematical notation x^y is neat for handwriting but awkward on typewriters and computers. So the programming languages have other ways of expressing exponentiation:

• x ↑ y: Algol, Commodore BASIC
• x ^ y: BASIC, Matlab, J programming language, Microsoft Excel, and most computer algebra systems
• x ** y: Fortran, Perl, Python (programming language), Ruby (programming language), Ada (programming language), FoxPro, SAS 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#
• (expt x y): Common Lisp

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.

		n
		1	2	3	4	5	6	7	8	9	10
k^	1	1	1	1	1	1	1	1	1	1	1	1
	2	2	4	8	16	32	64	128	256	512	1,024	2
	3	3	9	27	81	243	729	2,187	6,561	19,683	59,049	3
	4	4	16	64	256	1,024	4,096	16,384	65,536	262,144	1,048,576	4
	5	5	25	125	625	3,125	15,625	78,125	390,625	1,953,125	9,765,625	5
	6	6	36	216	1,296	7,776	46,656	279,936	1,679,616	10,077,696	60,466,176	6
	7	7	49	343	2,401	16,807	117,649	823,543	5,764,801	40,353,607	282,475,249	7
	8	8	64	512	4,096	32,768	262,144	2,097,152	16,777,216	134,217,728	1,073,741,824	8
	9	9	81	729	6,561	59,049	531,441	4,782,969	43,046,721	387,420,489	3,486,784,401	9
	10	10	100	1,000	10,000	100,000	1,000,000	10,000,000	100,000,000	1,000,000,000	10,000,000,000	10
		1	2	3	4	5	6	7	8	9	10
		n

Generalization

The next generalized operation after multiplication and exponentiation is sometimes called tetration; repeating this process leads to the Ackermann function.

See also

Wikimedia Commons has media related to:

Exponential function

• List of exponential topics
• Exponential growth
• Exponential decay
• Exponentiating by squaring
• Logarithm
• Modular exponentiation
• Addition chain exponentiation using an addition chain
• Tetration
• Unicode subscripts and superscripts
• Allometric function

External links

• sci.math FAQ: What is 0⁰?
• Introducing 0th power on PlanetMath
• Laws of Exponents with derivation and examples
• 1058 Powers of Two

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