rational exponent

A rational exponent is a fraction used as an exponent, and it bundles a root and a power into one symbol. The denominator gives a root and the numerator gives a power: b^(m/n) means take the nth root of b, then raise it to the m, or equivalently raise b to the m and then take the nth root.

Start with the simplest case: b^(1/n) is just the nth root of b. So 8^(1/3) = cbrt(8) = 2, and 9^(1/2) = sqrt(9) = 3. Then b^(m/n) = (b^(1/n))^m = (the nth root of b) raised to the m. For example 8^(2/3) = (cbrt(8))^2 = 2^2 = 4.

The real payoff is that all the ordinary exponent rules — adding, subtracting, multiplying exponents — now apply to roots, often making radical problems far simpler than juggling root signs. One caution: for even denominators the base should be nonnegative to stay in the real numbers, and you usually reduce the fraction first to avoid sign surprises.