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For even values of n , the expression 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 All the properties for integer exponents given in this section also apply to any rational exponent on a nonnegative real-number 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, |