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
- Definition:Power of Zero for the definition of $x^r$ where $x = 0$.
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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Laws of Exponents: $7.8$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.9$: Roots
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms: $(6)$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): exponent (index)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): exponent (index)
- 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $5$: Eternal Triangles: Logarithms
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 13$: Laws of Exponents: $13.8.$
- For a video presentation of the contents of this page, visit the Khan Academy.