1. Bell Work Simplify by adding like terms. 4 + 7mxy + 5 + 3yxm – 15

2. Answer: -6 + 10mxy

3. Lesson 19: Exponents, Powers of Negative Numbers, Roots, Evaluation of Powers

4. Exponents: Exponential Notation*: The general form of the expression is x, which indicates that x is be used as a factor n times and is read “x to the nth.” n

5. In this definition, the letter x represents a real number and is called the base of the expression. The letter n represents a positive integer and is called the exponent.

6. Power*: The value of an exponential expression. 2= (2)(2)(2)(2) = 16 The value of 2 used as a factor four times is 16. We say that the fourth power of 2 is 16. We could also say 2 raised to the 4 th power. 4

7. If something is raised to the 2 nd power we say it is squared. If something is raised to the 3 rd power we say it is cubed. Everything greater than 3 we say the nth power.

8. Powers of negative numbers: When a positive number is raised to a positive power, the result is always a positive number.

9. Example: Simplify 3 3 3 -3 2344

10. Answer: (3)(3) = 9 (3)(3)(3) = 27 (3)(3)(3)(3) = 81 Be careful because -3 means the opposite of 3 and not (-3) -(3)(3)(3)(3) = -81 4 4 4

11. When a negative number is raised to an even power, the result is always positive; and when a negative number is raised to an odd power, the result is always negative.

12. Example: Simplify (-3) (-3)(-3) -(-3) 234 4

13. Answer: (-3)(-3) = 9 (-3)(-3)(-3) = -27 (-3)(-3)(-3)(-3) = 81 -(-3)(-3)(-3)(-3) = -81

14. Example: Simplify -3 – (-3) + (-2) 322

15. Answer: -(3)(3)(3) – (-3)(-3) + (-2)(-2) = -27 – 9 + 4 = -32

16. Example: Simplify -2 – 4(-3) – 2(-2) – 2 232

17. Answer: -(2)(2) – 4[(-3)(-3)(-3)] – 2[(-2)(-2)] – 2 = -4 – 4(-27) – 2(4) – 2 = -4 + 108 – 8 – 2 = 94

18. Roots: If we use 3 as a factor twice, the result is 9. Thus, 3 is the positive square root of 9. we use a radical sign to indicate the root of a number. (3)(3) = 9 so √9 = 3

19. If we use 3 as a factor three times, the result is 27. thus, 3 is the positive cube root of 27. (3)(3)(3) = 27 so √27 = 3 3

20. If we use 3 as a factor four times, the result is 81. Thus, 3 is the positive fourth root of 81. (3)(3)(3)(3) = 81 so √81 = 3 4

21. If we use 3 as a factor five times, the result is 243. Thus, 3 is the positive fifth root of 243. (3)(3)(3)(3)(3) = 243 so √243 = 3 5

22. Because (-3)(-3) = +9, we say that - 3 is the negative square root of 9. Because (-3)(-3)(-3)(-3) equals +81, we say that -3 is the negative fourth root of 81.

23. If n is an even number, every positive real number has a positive nth root and a negative nth root. We use the radical sign to designate the positive even root. To designate a negative even root, we also use a minus sign.

24. Example: Simplify √9 -√9 √81-√81 4 4

25. Answer: √9 = 3 -√9 = -3 √81 = 3 -√81 = -3 4 4

26. The number under the radical sign is called the radicand, and the little number that designates the root is called the index.

27. Practice: √64 √16 √-27 -√81 43

28. Answer: √64 = 8 √16 = 2 √-27 = -3 -√81 = -9 4 3

29. Evaluation of Powers: Evaluate: yx m If y = 3, x = 4, and m = 2 (3)(4) (2) = (3)(16)(8) =384 23 23

30. We must be careful, however, when the expression contain minus signs or when some replacement values of the variables are negative numbers. If a = -2, what is the value of each of the following? a -a -a (-a) 2233

31. Answer: a = -2 a = (-2)(-2) = +4 -a = -(-2)(-2) = -4 -a = -(-2)(-2)(-2) = +8 (-a) = (-(-2))(-(-2))(-(-2)) = +8 2 2 3 3

32. Practice: Evaluate pm – z If p = 1, m = -4, z = -2 2 3

33. Answer: (1)(-4)(-4) – (-2)(-2)(-2) = 16 + 8 = 24

34. HW: Lesson 19 #1-30 Due Tomorrow