College Algebra Sixth Edition James Stewart  Lothar Redlin  Saleem Watson

1. College Algebra Sixth Edition James StewartLothar RedlinSaleem Watson

2. Prerequisites P

3. Rational Exponents and Radicals P.4

4. Exponents and Radicals • In this section, we learn to work with expressions that contain radicals or rational exponents.

5. Radicals

6. Radicals • We know what 2nmeans whenever n is an integer. • To give meaning to a power, such as 24/5, whose exponent is a rational number, we need to discuss radicals.

7. Radicals • The symbol √means: “the positive square root of.” • Thus,

8. Radicals • Since a =b2≥ 0, the symbol makes sense only when a ≥ 0. • For instance,

9. nth Root • Square roots are special cases of nth roots. • The nth root of x is the number that, when raised to the nth power, gives x.

10. nth Root—Definition • If n is any positive integer, then the principal nth root of a is defined as follows: • If n is even, we must have a ≥ 0 and b ≥ 0.

11. nth Roots • Thus,

12. nth Roots • However, , , and are not defined. • For instance, is not defined because the square of every real number is nonnegative.

13. nth Roots • Notice that • So, the equation is not always true. • It is true only when a ≥ 0.

14. nth Roots • However, we can always write • This last equation is true not only for square roots, but for any even root. • This and other rules used in working with nth roots are listed in the following box.

15. Properties of nth Roots In each property, we assume that all the given roots exist.

16. E.g. 1—Simplifying Expressions Involving nth Roots Simplify these expressions.

17. Example (a) E.g. 1—Expressions with nth Roots

18. Example (b) E.g. 1—Expressions with nth Roots

19. Combining Radicals • It is frequently useful to combine like radicals in an expression such as • This can be done using the Distributive Property. • Thus, • The next example further illustrates this process.

20. E.g. 2—Combining Radicals

21. E.g. 2—Combining Radicals

22. Rational Exponents

23. Rational Exponents • To define what is meant by a rational exponent or, equivalently, a fractional exponent such as a1/3, we need to use radicals.

24. Rational Exponents • To give meaning to the symbol a1/nin a way that is consistent with the Laws of Exponents, we would have to have: (a1/n)n =a(1/n)n =a1 = a • So, by the definition of nth root, • In general, we define rational exponents as follows.

25. Rational Exponent—Definition • For any rational exponent m/n in lowest terms, where m and n are integers and n > 0, we define • If n is even, we require that a ≥ 0. • With this definition, it can be proved that the Laws of Exponents also hold for rational exponents.

26. E.g. 3—Using the Definition of Rational Exponents

27. E.g. 3—Using the Definition of Rational Exponents

28. E.g. 4—Using the Laws of Exponents with Rational Exponents

29. E.g. 4—Using the Laws of Exponents with Rational Exponents

30. E.g. 4—Using the Laws of Exponents with Rational Exponents

31. E.g. 5—Simplifying by Writing Radicals as Rational Exponents

32. E.g. 5—Simplifying by Writing Radicals as Rational Exponents

33. E.g. 5—Writing Radicals as Rational Exponents

34. Rationalizing the Denominator; Standard Form

35. Rationalizing the Denominator • It is often useful to eliminate the radical in a denominator by multiplying both numerator and denominator by an appropriate expression. • This procedure is called rationalizing the denominator.

36. Rationalizing the Denominator • If the denominator is of the form , we multiply numerator and denominator by . • In doing this, we multiply the given quantity by 1. • So, we do not change its value.

37. Rationalizing the Denominator • For instance, • Note that the denominator in the last fraction contains no radical.

38. Rationalizing the Denominator • In general, if the denominator is of the form with m <nthen multiplying the numerator and denominator by will rationalize the denominator. • This is because (for a > 0)

39. E.g. 6—Rationalizing Denominators