Exponent Laws

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 and and real exponents and , the laws are

			(1)
			(2)
			(3)
			(4)
			(5)
			(6)
			(7)

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

(8)

and

(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 (Knuth 1992; Knuth 1997, p. 56).

The same rules also hold for nonzero complex and when and 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,

(10)

For , define using the principal value of the natural logarithm, with complex argument in . The precise nested-power relation is

(11)

where is the unique integer satisfying . Indeed, . Thus always holds for integer , and also whenever . Merely assuming that the variables are real is insufficient without sign restrictions on and . For example, for real number , which equals only for (Trott 2006).