
For even values of n , the expression is defined to be the positive nth root of a or the principal nth root of a. For example, denotes the positive second root, or square root, of a , while is the positive fourth root of a. When n is odd, there is only one n th root, which has the same sign as a. For example, , the cube root of a, has the same sign as a. By definition, if then . On a calculator, a number is raised to a power using a key labeled For example, to take the fourth root of 6 on a TI83 calculator, enter , to get the result 1.56508458. EXAMPLE 1 Calculations with Exponents
Rational ExponentsIn the following definition, the domain of an exponent is extended to include all rational numbers. DEFINITION OF For all real numbers a for which the indicated roots exist, and for any rational number m/n EXAMPLE 2 Calculations with Exponents NOTE could also be evaluated as but this is more difficult to perform without a calculator because it involves squaring 27, and then taking thecube root of this large number. On the other hand, when we evaluate it as we know that the cube root of 27 is 3 without using a calculator, and squaring 3 is easy. All the properties for integer exponents given in this section also apply to any rational exponent on a nonnegative realnumber base. EXAMPLE 3 Simplifying Exponential Expressions
In calculus, it is often necessary to factor expressions involving fractional exponents. EXAMPLE 4 Simplifying Exponential Expressions Factor out the smallest power of the variable, assuming all variables represent positive real numbers. Solution To check this result, multiply by Solution The smallest exponent here is 3. Since 3 is a common numerical factor, factor out Check by multiplying. The factored form can be written without negative exponents as . Solution There is a common factor of 2. Also, have a common factor. Always factor out the quantity to the smallest exponent. Here 1/2 < 1/2 so the common factor is and the factored form is 