Section 1: Exponents and Roots

Section 8.1 Exponents and Roots 761Hence, the approximate solutions are x ≈ −2.645751 or x ≈ 2.6457513.On the other hand, to find analytic solutions of x 2 = 7, we simply take plus orminus the square root of 7.x 2 = 7x = ± √ 7To compare these exact solutions with the approximate solutions found by using thegraphing calculator, use a calculator to compute ± √ 7, as shown in Figure 7.Figure 7. Approximating ± √ 7.Note that these approximations of − √ 7 and √ 7 agree quite nicely with the solutionsfound using the graphing calculator’s intersect utility and reported in Figure 6.Both − √ 7 and √ 7 are examples of irrational numbers, that is, numbers that cannotbe expressed in the form p/q, where p and q are integers.Rational ExponentsAs with the definition of negative and zero exponents, discussed earlier in this section,it turns out that rational exponents can be defined in such a way that the Laws ofExponents will still apply (and in fact, there’s only one way to do it).The third law gives us a hint on how to define rational exponents. For example,suppose that we want to define 2 1/3 . Then by the third law,( )2 1 3 13 = 2 3 ·3 = 2 1 = 2,so, by taking cube roots of both sides, we must define 2 1/3 by the formula 32 1 3 =3 √ 2.The same argument shows that if n is any odd positive integer, then 2 1/n must bedefined by the formula2 1 n =n √ 2.However, for an even integer n, there appears to be a choice. Suppose that we wantto define 2 1/2 . Then3 Recall that the equation x 3 = a has a unique solution x = 3√ a.Version: Fall 2007