Section P.2

1. Section P.2 Exponents and Radicals

2. Definition of Exponent • An exponent is the power p in an expression ap. 52 • The number 5 is the base. • The number 2 is the exponent. • The exponent is an instruction that tells us how many times to use the base in a multiplication.

3. Puzzler (-5)2 = -52? (-5)(-5) = -(5)(5) 25 = -25

4. 43 -34 (-2)5 (-3/4)2 =(4)(4)(4) = 64 =(-)(3)(3)(3)(3) = -81 =(-2)(-2)(-2)(-2)(-2)= -32 =(-3/4)(-3/4) = (9/16) Examples Which of these will be negative?

5. Multiplication with Exponents by Definition 3235 = (3)(3)(3)(3)(3)(3)(3) = 37 Note 2+5=7

6. Property 1 for Exponents • If a is any real number and r and s are integers, then To multiply two expressions with the same base, add exponents and use the common base.

7. Which one is it?

8. Which one is it?

9. Examples of Property 1

10. By the Definition of Exponents Notice that 5 – 3 = 2

11. Examples

12. Property 2 for Exponents • If a is any real number and r and s are integers, then To divide like bases subtract the exponents.

13. Negative Exponents Notice that 3 – 5 = -2

14. Property 3Definition of Negative Exponents • If nis a positive integer, then

15. Examples of Negative Exponents Notice that: Negative Exponents do not indicate negative numbers. Negative exponents do indicate Reciprocals.

16. Examples of Negative Exponents Notice that exponent does not touch the 3.

17. Zero as an Exponent

18. Zero to the Zero? Undefined STOP Zeros are not allowed in the denominator. So 00 is undefined.

19. Examples

20. Property 5 for Exponents • If a and b are any real number and r is an integer, then Distribute the exponent.

21. Examples of Property 5

22. Power to a Power by Definition (32)3 = ((3)(3))1((3)(3))1((3)(3))1 = 36 Note 3(2)=6

23. Property 6 for Exponents • If a is any real number and r and s are integers, then A power raised to another power is the base raised to the product of the powers.

24. Examples of Property 5 + 6

25. Examples of Property 6 One base, two exponents… multiply the exponents.

26. Definition of nth root of a number Let a and b be real numbers and let n ≥ 2 be a positive integer. If a = bn then b is the nth root of a. If n = 2, the root is a square root. If n = 3, the root is a cube root.

27. Property 2 for Radicals • The nth root of a product is the product of nth roots

28. Property 3 for Radicals • The nth root of a quotient is the quotient of the nth roots

29. For a radicand to come out of a radical the exponent must match the index. 10 4 8 3 3 2

30. Example 1

31. Example 2

32. Example 3

33. Simplified Form for Radical Expressions A radical expression is in simplified form if 1. All possible factors have been removed from the radical. None of the factors of the radicand can be written in powers greater than or equal to the index. 2. There are no radicals in the denominator. 3. The index of the radical is reduced.

34. Example 6

35. Example 7

36. Example 10

37. For something to go inside a radical the exponent must match the index.

38. Rationalize the denominator. This will always be a perfect square.

39. Rationalize the denominator. Often you will not need to write this step.

40. Rationalize the denominator. 5

41. Simplify numerator first, if possible

42. Simplify first then rationalize.

43. Reduce, simplify, rationalize

44. Product

45. Another product

46. Assume that r and t represent nonnegative real numbers.

47. Cube roots are a different story. Must have 3 of a kind

48. Cube Roots Must have 3 of a kind

49. More Cube Roots

50. Simplify first 3 2