P.4. Radicals and Rational Exponents
Thus far we have only discussed integer exponents. What about fractional (rational) exponents? What would we mean by 21/2, for instance? In order to define such expressions in a meaningful way, it is useful to first look at some useful properties of exponents. We state these properties without proof. The symbols a, b, and c stand for arbitrary nonzero real numbers with a positive, and b positive as well in (P.4.5).
- (P.4.1): ab+c = abac
- (P.4.2): a-b = 1/ab
- (P.4.3): ab-c = ab/ac
- (P.4.4): (ab)c = abc
- (P.4.5): acbc = (ab)c
Equation (P.4.4) is the one we will use to define rational exponents. To see this, consider the case a = 2, b = 1/2, and c = 2. According to (P.4.4), we see that (21/2)2 = 2(1/2)*2 = 21 = 2. Thus, 21/2 is the number whose square is equal to 2. In other words, 21/2 is the square root of 2, more commonly written as √2. More generally, for any positive number a, a1/2 is the square root of a, more commonly written as √a. Similarly, a1/3 is the cube root of a, i.e. the number whose cube is equal to a, which is also written as 3√a. To fully generalize, for every positive real number a and every positive integer k, a1/k is the kth root of a, i.e. the number whose kth power is equal to a. This is more commonly written as k√a. Such an expression is known as a radical expression and the symbol k√ is known as a radical.
Example 1: Simplify 21/2 and 41/2.
Solution: From the above discussion, we know that 21/2=√2, which cannot be further simplified. We also know that 41/2=√4=2 since 22=2*2=4.
Example 2: Simplify 31/3 and 32/3.
Solution: From the above discussion, we know that 31/3 = 3√3, which cannot be further simplified. We also know from (P.4.4) that
Example 3: Simplify 10001/3 and 10002/3.
Solution: We have 10001/3 = 3√1000 = 10, since 103 = 10*10*10 = 1000. We also know from (P.4.4) that