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Rational exponents

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

Main article: n-th root

An n-th root of a number a is a number x such that xⁿ = a.

If n is a positive integer and a is a positive real number, then there is exactly one positive real solution to xⁿ = a. This solution is called the principal n-th root of a. It is denoted ⁿ√a, where √  is the radical symbol; alternatively, it may be written a^1/n. For example: 4^1/2 = 2, 8^1/3 = 2,

When one speaks of the n-th root of a positive real number a, one usually means the principal n-th root.

If n is even, then xⁿ = a has two real solutions if a is positive, which are the positive and negative nth roots. The equation has no solution in real numbers if a is negative.

If n is odd, then xⁿ = a has one real solution. The solution is positive if a is positive and negative if a is negative.

Rational powers m/n, where m/n is in lowest terms, are positive if m is even, negative for negative a if m and n are odd, and can be either sign if a is positive and n is even. (−27)^1/3 = −3, (−27)^2/3 = 9, and 4^3/2 has two roots 8 and −8. Since there is no real number x such that x² = −1, the definition of a^m/n when a is negative and n is even must use the imaginary unit i, as described more fully in the section Powers of complex numbers.

A power of a positive real number a with a rational exponent m/n in lowest terms satisfies

$a^{m/n}=\left(a^{m}\right)^{1/n}={\sqrt[{n}]{a^{m}}}$

where m is an integer and n is a positive integer.

Care needs to be taken when applying the power law identities with negative nth roots. For instance, −27 = (−27)^{((2/3)×(3/2))} = ((−27)^2/3)^3/2 = 9^3/2 = 27 is clearly wrong. The problem here occurs in taking the positive square root rather than the negative one at the last step, but in general the same sorts of problems occur as described for complex numbers in the section Failure of power and logarithm identities.

Real powers of positive numbers

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 roots, 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 for integer exponents are true for positive real numbers with noninteger exponents as well.

Extension of rational exponents

Since any irrational number can be approximated by rational numbers, exponentiation to an arbitrary real exponent x can be defined by continuity with the rule

$b^{x}=\lim _{r\to x}b^{r}\quad (r\in \mathbb {Q} ,\,x\in \mathbb {R} ),$

where the limit as r gets close to x is taken only over rational values of r.

For example, if

$x\approx 1.732$

then

$5^{x}\approx 5^{1.732}=5^{433/250}={\sqrt[{250}]{5^{433}}}\approx 16.241.$

Exponentiation by a real power is normally accomplished using logarithms instead of using limits of rational powers.

Powers of e

Main article: Exponential function

The important mathematical constant e, sometimes called Euler's number, is approximately equal to 2.718 and is the base of the natural logarithm. It provides a path for defining exponentiation with noninteger exponents. It is defined as the following limit where the power goes to infinity as the base tends to one:

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

The exponential function, defined by

$e^{x}=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n},$

has the x written as a power as it satisfies the basic exponential identity

$e^{x+y}=e^{x}\cdot e^{y}.$

The exponential function is defined for all integer, fractional, real, and complex values of x. It can even be used to extend exponentiation to some nonnumerical entities such as square matrices; however, the exponential identity only holds when x and y commute.

A short proof that e to a positive integer power k is the same as e^k is:

${\begin{aligned}(e)^{k}&=\left[\lim _{n\rightarrow \infty }\left(1+{\frac {1}{n}}\right)^{n}\right]^{k}=\lim _{n\rightarrow \infty }\left[\left(1+{\frac {1}{n}}\right)^{n}\right]^{k}\\&=\lim _{n\rightarrow \infty }\left(1+{\frac {k}{n\cdot k}}\right)^{n\cdot k}=\lim _{n\cdot k\rightarrow \infty }\left(1+{\frac {k}{n\cdot k}}\right)^{n\cdot k}\\&=\lim _{m\rightarrow \infty }\left(1+{\frac {k}{m}}\right)^{m}=e^{k}\end{aligned}}$

This proof shows also that e^x+y satisfies the exponential identity when x and y are positive integers. These results are in fact generally true for all numbers, not just for the positive integers.

Real exponents

The natural logarithm ln(x) is the inverse of the exponential function e^x. It is defined for b > 0, and satisfies

$b=e^{\ln b}.\,$

If b^x is to preserve the logarithm and exponent rules, then one must have

$b^{x}=(e^{\ln b})^{x}=e^{x\cdot \ln b}$

for each real number x.

This can be used as an alternative definition of the real number power b^x and 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.

Definitions

$b^{n}=\underbrace {b\times \cdots \times b} _{n}$ (b positive, integer exponent n)

$b^{0}=1$ (exponent zero)

$b^{-n}={1}/{b^{n}}$ (negative integer exponent)

$b^{\frac {m}{n}}=\left(b^{m}\right)^{\frac {1}{n}}={\sqrt[{n}]{b^{m}}}$ (rational exponent m/n)

$b^{x}=(e^{\ln b})^{x}=e^{x\cdot \ln b}$ (real exponent x)

$b^{z}=e^{z\cdot \ln b}=e^{(x+iy)\cdot \ln b}$ (complex exponent z=x+iy)

$z^{n}=\underbrace {z\times \cdots \times z} _{n}$ (complex base z, integer exponent n)

$(-b)^{n}=(-1)^{n}\cdot b^{n}$ (special case: negative base)

$(iy)^{n}=i^{n}\cdot y^{n}$ (special case: imaginary base)

$w^{z}=e^{z\log w}$ (complex base and exponent, ambiguous in the same sense that log w is)

$w^{z}=e^{z\log w}=e^{z(\log r+i\theta )}$ (with $w=re^{i\theta }$ in polar form)