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Exponent Laws

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

For positive real numbers x and y and real exponents m and n, the laws are

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).

The same rules also hold for nonzero complex x and y when m and n are integers. For arbitrary complex exponents, identities involving nested powers or powers of products can fail when each power uses its principal value. For example,

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

For z!=0, define z^a=exp(alnz) using the principal value of the natural logarithm, with complex argument in (-pi,pi]. The precise nested-power relation is

(z^a)^b=z^(ab)e^(-2piikb),
(11)

where k is the unique integer satisfying -pi<Im(alnz)-2pik<=pi. Indeed, ln(z^a)=alnz-2piik. Thus (z^a)^b=z^(ab) always holds for integer b, and also whenever k=0. Merely assuming that the variables are real is insufficient without sign restrictions on x and y. For example, sqrt(x^2)=|x| for real number x, which equals x only for x>=0 (Trott 2006).

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.National Institute of Standards and Technology. "The Logarithm" and "Powers." §4.2(i) and 4.2(iv) in Digital Library of Mathematical Functions. https://dlmf.nist.gov/4.2.Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, sec. 2.2.6, 2004. https://www.mathematicaguidebooks.org/.Trott, M. "Operations on Trigonometric Expressions." §1.4 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. https://doi.org/10.1007/0-387-28815-5. https://www.mathematicaguidebooks.org/.

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Exponent Laws

Cite this as:

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

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