Exponentiation - Rational Exponents

Rational Exponents

An n-th root of a number b is a number x such that xn = b.

If b is a positive real number and n is a positive integer, then there is exactly one positive real solution to xn = b. This solution is called the principal n-th root of b. It is denoted n√b, where √  is the radical symbol; alternatively, it may be written b1/n. For example: 41/2 = 2, 81/3 = 2,

When one speaks of the n-th root of a positive real number b, one usually means the principal n-th root.

If n is even, then xn = b has two real solutions if b is positive, which are the positive and negative nth roots. The equation has no solution in real numbers if b is negative.

If n is odd, then xn = b has one real solution. The solution is positive if b is positive and negative if b is negative.

Rational powers m/n, where m/n is in lowest terms, are positive if m is even, negative for negative b if m and n are odd, and can be either sign if b is positive and n is even. (−27)1/3 = −3, (−27)2/3 = 9, and 43/2 has two roots 8 and −8. Since there is no real number x such that x2 = −1, the definition of bm/n when b is negative and n is even must use the imaginary unit i, as described more fully in the section Powers of complex numbers.

A power of a positive real number b with a rational exponent m/n in lowest terms satisfies

where m is an integer and n is a positive integer.

Care needs to be taken when applying the power law identities with negative nth roots. For instance, −27 = (−27)((2/3)⋅(3/2)) = ((−27)2/3)3/2 = 93/2 = 27 is clearly wrong. The problem here occurs in taking the positive square root rather than the negative one at the last step, but in general the same sorts of problems occur as described for complex numbers in the section Failure of power and logarithm identities.