structural:summarization, information:extraction
Rational exponents

If x is a nonnegative real number, and n is a positive integer, x 1 / n {\displaystyle x^{1/n}} or x n {\displaystyle {\sqrt{x}}} denotes the unique positive real n th root of x, that is, the unique positive real number y such that y n = x. {\displaystyle y^{n}=x.}

If x is a positive real number, and p q {\displaystyle {\frac {p}{q}}} is a rational number, with p and q > 0 integers, then x p / q {\textstyle x^{p/q}} is defined as

The equality on the right may be derived by setting y = x 1 q, {\displaystyle y=x^{\frac {1}{q}},} and writing (x 1 q) p = y p = ((y p) q) 1 q = ((y q) p) 1 q = (x p) 1 q. {\displaystyle (x^{\frac {1}{q}})^{p}=y^{p}=\left((y^{p})^{q}\right)^{\frac {1}{q}}=\left((y^{q})^{p}\right)^{\frac {1}{q}}=(x^{p})^{\frac {1}{q}}.}

If r is a positive rational number, 0 = 0, by definition.

All these definitions are required for extending the identity (x r) s = x r s {\displaystyle (x^{r})^{s}=x^{rs}} to rational exponents.

On the other hand, there are problems with the extension of these definitions to bases that are not positive real numbers. For example, a negative real number has a real n th root, which is negative, if n is odd, and no real root if n is even. In the latter case, whichever complex n th root one chooses for x 1 n, {\displaystyle x^{\frac {1}{n}},} the identity (x a) b = x a b {\displaystyle (x^{a})^{b}=x^{ab}} cannot be satisfied. For example,

See § Real exponents and § Non-integer powers of complex numbers for details on the way these problems may be handled.