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8th Grade Slides. Exponents. Exponents. Key Skill: Calculate values of expressions with positive and negative exponents. Quick Review. Exponent. Base. Exponent Review. Repeated multiplication:. Exponent Review. Any number to the first power is?. Exponent Review.

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2. Exponents • Key Skill: Calculate values of expressions with positive and negative exponents.

3. Quick Review Exponent Base

4. Exponent Review • Repeated multiplication:

5. Exponent Review • Any number to the first power is?

6. Exponent Review • Any number to the first power is itself • So • And

7. Exponent Review • Any non-zero number to the power of 0 is?

8. Exponent Review • Any non-zero number to the power of 0 is 1. • So, • and (assuming x does not = 0),

9. Question • What is

10. Question • What is • Is it the same as

11. Odd and Even Exponents • Will a negative number raised to a power always be positive?

12. Odd and Even Exponents • How do we know whether a negative number raised to a particular power will be positive or negative?

13. Odd and Even Exponents • How do we know whether a negative number raised to a particular power will be positive or negative? Negative base with an odd exponent results in a NEGATIVE answer Negative base with an even exponent results in a POSITIVE answer

14. Decimals and Fractions • Is this the same as

15. Decimals and Fractions • Is this the same as No!

16. Negative Exponents • Look at this progression of numbers: 33 = 32 = 31 = 30 = 3-1 = 3-2 = 3-3 =

17. Negative Exponents • Look at this progression of numbers: 33 = 27 32 = 9 31 = 3 30 = 1 3-1 = 1/3 3-2 = 1/9 3-3 = 1/27

18. Negative Exponents • Rule:

19. Negative Exponents • The reverse is also true:

20. Negative Exponents • We generally do NOT want negative exponents in our answer (unless we are using Scientific Notation). • We move bases with negative exponents to the denominator where they will turn positive. • If a negative exponent is found in a denominator, we move it to the numerator.

21. Method Step 1: Take the RECIPROCAL of the base. What does RECIPROCAL MEAN? Step 2: The exponent moves with the base, but the negative sign in the exponent disappears.

22. Examples

23. Examples

24. Exception • Numbers generally get LARGER when we square them. What kind of numbers get SMALLER when we square them?

25. Exception • Numbers generally get LARGER when we square them. What kind of numbers get SMALLER when we square them? • Numbers between 0 and 1 (fractions and decimals)

26. Classwork • Pages

27. Exponent Laws • Key Skill: Apply the Laws of Exponents to simplify expressions.

28. Exponents • Reminders: • Anything raised to the power of 1 is itself. • Any non-zero number raised to the power of 0 is 1. • When we see we use PEMDAS to determine what to do first. • In this case, raise ‘x’ to the 3rd power, THEN multiply by 4

29. PEMDAS Parentheses ( ) Exponents 3 Multiplication x, Division ÷ Addition + Subtraction __

30. Please Excuse My Dear Aunt Sally • Parentheses ( ) • Exponents 3 • Multiplication x, • Division ÷ • Addition + • Subtraction __

31. Product Law of Exponents • How would we multiply:

32. Product Law of Exponents • How would we multiply: • We could rewrite it as: (4 x 4 x 4) x (4 x 4)

33. Product Law of Exponents • How would we multiply: • We could rewrite it as: (4 x 4 x 4) x (4 x 4) • Or with an exponent: • Note that we did NOT multiply the exponents, we ADDED them!

34. Another Example • How would we multiply:

35. Another Example • How would we multiply: • We would rewrite the equation as: • Or more simply as: 32 • Again, we ADDED the exponents

36. Example with a variable • What is:

37. Example with a variable • What is: • Rewrite as: • Or more simply as:

38. Product Law of Exponents

39. Contra-Example • Does the Product Law of Exponents apply here?

40. Contra-Example • Does the Product Law of Exponents apply here? • No! We do NOT have a common base or a common exponent.

41. Product Law of Exponents • What about:

42. Product Law of Exponents • What about: • Does it matter what order we write the factors?

43. Product Law of Exponents • What about: • Does it matter what order we write the factors? • How about:

44. Product Law of Exponents • What about: • Does it matter what order we write the factors? • How about:

45. More Examples

46. More Examples

47. The Other Product Law • How would we work with: • What’s different/what’s the same?

48. The Other Product Law • How would we work with: • We can rewrite as: (4 x 4) x (3 x 3) • Or as (4 x 3) x (4 x 3) because order doesn’t matter when we multiply

49. The Other Product Law • How would we work with: • We can rewrite as: (4 x 4) x (3 x 3) • Or as (4 x 3) x (4 x 3) because order doesn’t matter when we multiply • Now we can rewrite as

50. The Other Product Law • How would we work with: • What’s different/what’s the same?

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