Definition:Power (Algebra)/Rational Number

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Definition

Let $x \in \R$ be a real number such that $x > 0$.

Let $m \in \Z$ be an integer.

Let $y = \sqrt [m] x$ be the $m$th root of $x$.


Then we can write $y = x^{1/m}$ which means the same thing as $y = \sqrt [m] x$.


Thus we can define the power to a positive rational number:

Let $r = \dfrac p q \in \Q$ be a positive rational number where $p \in \Z_{\ge 0}, q \in \Z_{> 0}$.

Then $x^r$ is defined as:

$x^r = x^{p/q} = \paren {\sqrt [q] x}^p = \sqrt [q] {\paren {x^p} }$.


When $r = \dfrac {-p} q \in \Q: r < 0$ we define:

$x^r = x^{-p/q} = \dfrac 1 {x^{p/q}}$ analogously for the negative integer definition.


Examples

Fractional Power: $8^{2/3}$

$8^{2/3} = 4$


Fractional Power: $27^{-4/3}$

$27^{-4/3} = \dfrac 1 {81}$


Fractional Power: $32^{6/5}$

$32^{6/5} = 64$


Fractional Power: $0 \cdotp 125^{-2/3}$

$0 \cdotp 125^{-2/3} = 4$


Also see


Historical Note

The definition:

$x^r = x^{p/q} = \left({\sqrt [q] x}\right)^p = \sqrt [q] {\left({x^p}\right)}$

is due to Nicole Oresme circa $1360$.


The concept of a fractional exponent was reintroduced by John Wallis in the $17$th century.


Sources