HOME David Terr Ph.D. Math, UC Berkeley 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 32/3=32*1/3=(32)1/3=91/3=3√9.   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 10002/3=10001/3*2=(10001/3)2=102=100.   >> P.5. Polynomials Copyright © 2007-2009 - MathAmazement.com. All Rights Reserved.