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Exponent Laws

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The exponent laws, also called the laws of indices (Higgens 1998) or power rules (Derbyshire 2004, p. 65), are the rules governing the combination of exponents (powers).

The laws are given by

x^m·x^n=x^(m+n)
(1)
(x^m)/(x^n)=x^(m-n)
(2)
(x^m)^n=x^(mn)
(3)
(xy)^m=x^my^m
(4)
(x/y)^n=(x^n)/(y^n)
(5)
x^(-n)=1/(x^n)
(6)
(x/y)^(-n)=(y/x)^n,
(7)

where quantities in the denominator are taken to be nonzero. Special cases include

x^1=x
(8)

and

x^0=1
(9)

for x!=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 (Knuth 1992; Knuth 1997, p. 56).

Note that these rules apply in general only to real quantities, and can give manifestly wrong results if they are blindly applied to complex quantities. For example,

(i-1)^(2i)!=[(i-1)^2]^i.
(10)

In particular, for complex z and real a,

z^(ia)=e^(-aarg(z))(z^2)^(ia/2),
(11)

where arg(z) is the complex argument.

See also

Complex Exponentiation, Exponent, Exponential Function, Power

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References

Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004.Higgins, P. M. Mathematics for the Curious. Oxford, England: Oxford University Press, 1998.Knuth, D. E. "Two Notes on Notation." Amer. Math. Monthly 99, 403-422, 1992.Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison-Wesley, p. 56, 1997.Krantz, S. G. "Laws of Exponentiation." §1.2.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 8, 1999.

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Exponent Laws

Cite this as:

Weisstein, Eric W. "Exponent Laws." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ExponentLaws.html

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