Rational Exponents, Radicals, and Complex Numbers

1. Chapter 7 Rational Exponents, Radicals, and Complex Numbers

2. Radicals and Radical Functions § 7.1

3. Square Roots Opposite of squaring a number is taking the square root of a number. A number b is a square root of a number a if b2 = a. In order to find a square root of a, you need a # that, when squared, equals a.

4. Principal Square Roots is the negative square root of a. Principal and Negative Square Roots If a is a nonnegative number, then is the principal or nonnegative squareroot of a

5. Radicands Radical expression is an expression containing a radical sign. Radicand is the expression under a radical sign. Note that if the radicand of a square root is a negative number, the radical is NOT a real number.

6. Radicands Example:

7. Perfect Squares Square roots of perfect square radicands simplify to rational numbers (numbers that can be written as a quotient of integers). Square roots of numbers that are not perfect squares (like 7, 10, etc.) are irrationalnumbers. IF REQUESTED, you can find a decimal approximation for these irrational numbers. Otherwise, leave them in radical form.

8. Perfect Square Roots Radicands might also contain variables and powers of variables. To avoid negative radicands, assume for this chapter that if a variable appears in the radicand, it represents positive numbers only. Example:

9. Cube Roots Cube Root The cube root of a real number a is written as

10. Cube Roots Example:

11. nth Roots Other roots can be found, as well. The nth root of a is defined as If the index, n, is even, the root is NOT a real number when a is negative. If the index is odd, the root will be a real number.

12. nth Roots Example: Simplify the following.

13. nth Roots Example: Simplify the following. Assume that all variables represent positive numbers.

14. If the index of the root is even, then the notation represents a positive number. nth Roots But we may not know whether the variable a is a positive or negative value. Since the positive square root must indeed be positive, we might have to use absolute value signs to guarantee the answer is positive.

15. Finding nth Roots If n is an even positive integer, then If n is an odd positive integer, then

16. Finding nth Roots Simplify the following. If we know for sure that the variables represent positive numbers, we can write our result without the absolute value sign.

17. Finding nth Roots Example: Simplify the following. Since the index is odd, we don’t have to force the negative root to be a negative number. If a or b is negative (and thus changes the sign of the answer), that’s okay.

18. Evaluating Rational Functions Find the value We can also use function notation to represent rational functions. For example, Evaluating a rational function for a particular value involves replacing the value for the variable(s) involved. Example:

19. Root Functions Since every value of x that is substituted into the equation produces a unique value of y, the root relation actually represents a function. The domain of the root function when the index is even, is all nonnegative numbers. The domain of the root function when the index is odd, is the set of all real numbers.

20. Root Functions We have previously worked with graphing basic forms of functions so that you have some familiarity with their general shape. You should have a basic familiarity with root functions, as well.

21. Graph y 6 4 2 xy (6, ) (4, 2) (2, ) 2 x (1, 1) 1 1 (0, 0) 0 0 Graphs of Root Functions Example:

22. Graph y xy 8 2 4 (8, 2) (4, ) x (1, 1) 1 -1 -1 1 (-1, -1) (0, 0) 0 0 (-4, ) (-8, -2) -4 -8 -2 Graphs of Root Functions Example:

23. § 7.2 Rational Exponents

24. Exponents with Rational Numbers So far, we have only worked with integer exponents. In this section, we extend exponents to rational numbers as a shorthand notation when using radicals. The same rules for working with exponents will still apply.

25. Understanding a1/n If n is a positive integer greater than 1 and is a real number, then Recall that a cube root is defined so that However, if we let b = a1/3, then Since both values of b give us the same a,

26. Using Radical Notation Example: Use radical notation to write the following. Simplify if possible.

27. Understanding am/n as long as is a real number If m and n are positive integers greater than 1 with m/n in lowest terms, then

28. Using Radical Notation Example: Use radical notation to write the following. Simplify if possible.

29. Understanding am/n as long as a-m/n is a nonzero real number.

30. Using Radical Notation Example: Use radical notation to write the following. Simplify if possible.

31. Using Rules for Exponents Example: Use properties of exponents to simplify the following. Write results with only positive exponents.

32. Using Rational Exponents Example: Use rational exponents to write as a single radical.

33. § 7.3 Simplifying Radical Expressions

34. If and are real numbers, then Product Rule for Radicals Product Rule for Radicals

35. Simplifying Radicals Example: Simplify the following radical expressions. No perfect square factor, so the radical is already simplified.

36. Simplifying Radicals Example: Simplify the following radical expressions.

37. If and are real numbers, Quotient Rule Radicals Quotient Rule for Radicals

38. Simplifying Radicals Example: Simplify the following radical expressions.

39. The Distance Formula Distance Formula The distance d between two points (x1,y1) and (x2,y2) is given by

40. The Distance Formula Example: Find the distance between (5, 8) and (2, 2).

41. The Midpoint Formula Midpoint Formula The midpoint of the line segment whose endpoints are (x1,y1) and (x2,y2) is the point with coordinates

42. The Midpoint Formula Example: Find the midpoint of the line segment that joins points P(5, 8) and P(2, 2).

43. § 7.4 Adding, Subtracting, and Multiplying Radical Expressions

44. Sums and Differences Rules in the previous section allowed us to split radicals that had a radicand which was a product or a quotient. We can NOT split sums or differences.

45. Like Radicals In previous chapters, we’ve discussed the concept of “like” terms. These are terms with the same variables raised to the same powers. They can be combined through addition and subtraction. Similarly, we can work with the concept of “like” radicals to combine radicals with the same radicand. Like radicals are radicals with the same index and the same radicand. Like radicals can also be combined with addition or subtraction by using the distributive property.

46. Adding and Subtracting Radical Expressions Example: Can not simplify Can not simplify

47. Adding and Subtracting Radical Expressions Example: Simplify the following radical expression.

48. Adding and Subtracting Radical Expressions Example: Simplify the following radical expression.

49. Adding and Subtracting Radical Expressions Example: Simplify the following radical expression. Assume that variables represent positive real numbers.

50. Multiplying and Dividing Radical Expressions If and are real numbers,