SECTION 5.1 RATIONAL EXPONENTS

nth Root Theorem

If x is a real number and n is a natural number, then

For x < 0:    If n is odd, x has exactly one real nth root. This nth root is negative.
                    If n is even, x has no real nth root.

For x = 0:    Zero is the only nth root of zero for all natural numbers n.

For x > 0:    If n is odd, x has exactly one ral nth root. This nth root is positive.
                    If n is even, x has two real nth roots. One nth root is negative, and the other (its opposite) is
                    positive.

EXAMPLES: Square root(s) of 49: 7 and -7

                        Square root(s) of -25: none

                        Cube root(s) of 27: 3

                        Cube root(s) of -8: -2

                        Fifth root(s) of 0: 0

The Prinicipal nth root

The principal nth root of the real number x is denoted by either    

    For x < 0: If n is odd, the principal root is negative.
                        If n is even, there is no real nth root.

    For x = 0: The principal nth root is 0.

    For x > 0: The principal root is positive for all natural numbers n.

(The 4 is all that is being raised to the one-half power.)

Rational Exponents:

For a real number x and natural numbers m and n,

, if is a real number.