The laws governing the combination of Exponents (Powers) are
![$\displaystyle x^m\cdot x^n$](e_2648.gif) |
![$\textstyle =$](e_13.gif) |
![$\displaystyle x^{m+n}$](e_2649.gif) |
(1) |
![$\displaystyle {x^m\over x^n}$](e_2650.gif) |
![$\textstyle =$](e_13.gif) |
![$\displaystyle x^{m-n}$](e_2651.gif) |
(2) |
![$\displaystyle (x^m)^n$](e_2652.gif) |
![$\textstyle =$](e_13.gif) |
![$\displaystyle x^{mn}$](e_2653.gif) |
(3) |
![$\displaystyle (xy)^m$](e_2654.gif) |
![$\textstyle =$](e_13.gif) |
![$\displaystyle x^m y^m$](e_2655.gif) |
(4) |
![$\displaystyle \left({x\over y}\right)^n$](e_2656.gif) |
![$\textstyle =$](e_13.gif) |
![$\displaystyle {x^n\over y^n}$](e_2657.gif) |
(5) |
![$\displaystyle x^{-n}$](e_2658.gif) |
![$\textstyle =$](e_13.gif) |
![$\displaystyle {1\over x^n}$](e_2659.gif) |
(6) |
![$\displaystyle \left({x\over y}\right)^{-n}$](e_2660.gif) |
![$\textstyle =$](e_13.gif) |
![$\displaystyle \left({y\over x}\right)^n,$](e_2661.gif) |
(7) |
where quantities in the Denominator are taken to be nonzero. Special cases include
![\begin{displaymath}
x^1=x
\end{displaymath}](e_2662.gif) |
(8) |
and
![\begin{displaymath}
x^0=1
\end{displaymath}](e_2663.gif) |
(9) |
for
. The definition
is sometimes used to simplify formulas, but it should be kept in mind
that this equality is a definition and not a fundamental mathematical truth.
See also Exponent, Power
© 1996-9 Eric W. Weisstein
1999-05-25