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Exponent Laws

The laws governing the combination of Exponents (Powers) are

$\displaystyle x^m\cdot x^n$ $\textstyle =$ $\displaystyle x^{m+n}$ (1)
$\displaystyle {x^m\over x^n}$ $\textstyle =$ $\displaystyle x^{m-n}$ (2)
$\displaystyle (x^m)^n$ $\textstyle =$ $\displaystyle x^{mn}$ (3)
$\displaystyle (xy)^m$ $\textstyle =$ $\displaystyle x^m y^m$ (4)
$\displaystyle \left({x\over y}\right)^n$ $\textstyle =$ $\displaystyle {x^n\over y^n}$ (5)
$\displaystyle x^{-n}$ $\textstyle =$ $\displaystyle {1\over x^n}$ (6)
$\displaystyle \left({x\over y}\right)^{-n}$ $\textstyle =$ $\displaystyle \left({y\over x}\right)^n,$ (7)

where quantities in the Denominator are taken to be nonzero. Special cases include
\end{displaymath} (8)

\end{displaymath} (9)

for $x\not=0$. The definition $0^0=1$ 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