Rational Exponents

Rational Exponents

Expressions like  a^{1/2 ,} 5^-1/4 ,and (2y)^4/5 have not yet been defined. We will define

such expressions so that the general properties of exponents hold.

Consider a^1/2.a^1/2. If we want to multiply by adding exponents, it must

follow that a^1/2 . a^1/2 = a^{1/2 +1/2,} or a¹ Thus we should define a^1/2 to be a

square root of a. Similarly, a^1/3.a^1/3 a^1/3 = a^1/3+1/3+1/3,or  a¹,  so a^1/3

should be defined to mean .

 

a^1/n

For any nonnegative real number a and any natural number index n

(n ≠1)

a^1/n  means   (the nonnegative nth root of a).

a^m/n

For any natural numbers m and n (n ≠1)and any nonnegative real

number a,

a^m/n means   , or