Fractional exponents do "jive" with previously developed mathematical ideas. As a simple example, ÷4 can be written as 41/2. We know that (÷4)(÷4) = 4. What about (41/2)(41/2)? Well, in previous use of positive integer exponents, we learned, for instance, that x2x7 = x9. That is, we added the exponents in this example of products when the two numbers had the same base. If we do the same with the fractional exponents, we would obtain (41/2)(41/2) = 41/2+1/2 = 41 = 4. Hence, we do get "consistency" if we write ÷4 = 41/2 .
This definition is important: If m and n are positive integers, then
Some simple examples:
Fractional exponents offer some tremendous advantages when one works in higher branches of mathematics, such as calculus.