1. Math 71B
7.2 – Rational Exponents

2. Let’s define what it means to have a rational # as an exponent: 𝒂 𝟏 𝒏 = 𝒏 𝒂 Ex = 81 =𝟗 −64 1/3 = 3 −64 =−𝟒 7 𝑥 2 𝑦 5𝑧 = 𝒙 𝟐 𝒚 𝟓𝒛 𝟏 𝟕 (we’ll talk about how to simplify this further later on…)

3. Let’s define what it means to have a rational # as an exponent: 𝒂 𝟏 𝒏 = 𝒏 𝒂 Ex = 81 =𝟗 −64 1/3 = 3 −64 =−𝟒 7 𝑥 2 𝑦 5𝑧 = 𝒙 𝟐 𝒚 𝟓𝒛 𝟏 𝟕 (we’ll talk about how to simplify this further later on…)

4. Let’s define what it means to have a rational # as an exponent: 𝒂 𝟏 𝒏 = 𝒏 𝒂 Ex = 81 =𝟗 −64 1/3 = 3 −64 =−𝟒 7 𝑥 2 𝑦 5𝑧 = 𝒙 𝟐 𝒚 𝟓𝒛 𝟏 𝟕 (we’ll talk about how to simplify this further later on…)

5. Let’s define what it means to have a rational # as an exponent: 𝒂 𝟏 𝒏 = 𝒏 𝒂 Ex = 81 =𝟗 −64 1/3 = 3 −64 =−𝟒 7 𝑥 2 𝑦 5𝑧 = 𝒙 𝟐 𝒚 𝟓𝒛 𝟏 𝟕 (we’ll talk about how to simplify this further later on…)

6. Let’s define what it means to have a rational # as an exponent: 𝒂 𝟏 𝒏 = 𝒏 𝒂 Ex = 81 =𝟗 −64 1/3 = 3 −64 =−𝟒 7 𝑥 2 𝑦 5𝑧 = 𝒙 𝟐 𝒚 𝟓𝒛 𝟏 𝟕 (we’ll talk about how to simplify this further later on…)

7. Let’s define what it means to have a rational # as an exponent: 𝒂 𝟏 𝒏 = 𝒏 𝒂 Ex = 81 =𝟗 −64 1/3 = 3 −64 =−𝟒 7 𝑥 2 𝑦 5𝑧 = 𝒙 𝟐 𝒚 𝟓𝒛 𝟏 𝟕 (we’ll talk about how to simplify this further later on…)

8. Note: We can write 𝑎 2 3 in a couple of different ways: 𝑎 2 3 = 𝑎 = 𝟑 𝒂 𝟐 or 𝑎 2 3 = 𝑎 = 𝟑 𝒂 𝟐

9. Note: We can write 𝑎 2 3 in a couple of different ways: 𝑎 2 3 = 𝑎 = 𝟑 𝒂 𝟐 or 𝑎 2 3 = 𝑎 = 𝟑 𝒂 𝟐

10. Ex 2. Use radical notation to rewrite each expression and simplify

11. Ex 2. Use radical notation to rewrite each expression and simplify

12. Ex 2. Use radical notation to rewrite each expression and simplify

13. Ex 2. Use radical notation to rewrite each expression and simplify

14. Ex 2. Use radical notation to rewrite each expression and simplify

15. Ex 2. Use radical notation to rewrite each expression and simplify

16. Ex 3. Rewrite with rational exponents: 3 5 4 = 𝟓 𝟒 𝟑 4 11 𝑥 2 𝑦 9 = 𝟏𝟏 𝒙 𝟐 𝒚 𝟗 𝟒

17. Ex 3. Rewrite with rational exponents: 3 5 4 = 𝟓 𝟒 𝟑 4 11 𝑥 2 𝑦 9 = 𝟏𝟏 𝒙 𝟐 𝒚 𝟗 𝟒

18. Ex 3. Rewrite with rational exponents: 3 5 4 = 𝟓 𝟒 𝟑 4 11 𝑥 2 𝑦 9 = 𝟏𝟏 𝒙 𝟐 𝒚 𝟗 𝟒

19. Radical
Notation
Rational
Exponent
7 𝑥 𝑥
3 𝑥 5
𝑥 3
𝑥 1 8
𝑥 1 2
𝑥 5 6
𝑥 7 2

20. Radical
Notation
Rational
Exponent
7 𝑥
𝒙 𝟏 𝟕 𝑥
3 𝑥 5
𝑥 3
𝑥 1 8
𝑥 1 2
𝑥 5 6
𝑥 7 2

21. Radical
Notation
Rational
Exponent
7 𝑥
𝒙 𝟏 𝟕 𝑥
𝒙 𝟏 𝟐
3 𝑥 5
𝑥 3
𝑥 1 8
𝑥 1 2
𝑥 5 6
𝑥 7 2

22. Radical
Notation
Rational
Exponent
7 𝑥
𝒙 𝟏 𝟕 𝑥
𝒙 𝟏 𝟐
3 𝑥 5
𝒙 𝟓 𝟑
𝑥 3
𝑥 1 8
𝑥 1 2
𝑥 5 6
𝑥 7 2

23. Radical
Notation
Rational
Exponent
7 𝑥
𝒙 𝟏 𝟕 𝑥
𝒙 𝟏 𝟐
3 𝑥 5
𝒙 𝟓 𝟑
𝑥 3
𝒙 𝟑 𝟐
𝑥 1 8
𝑥 1 2
𝑥 5 6
𝑥 7 2

24. Radical
Notation
Rational
Exponent
7 𝑥
𝒙 𝟏 𝟕 𝑥
𝒙 𝟏 𝟐
3 𝑥 5
𝒙 𝟓 𝟑
𝑥 3
𝒙 𝟑 𝟐
𝟖 𝒙
𝑥 1 8
𝑥 1 2
𝑥 5 6
𝑥 7 2

25. Radical
Notation
Rational
Exponent
7 𝑥
𝒙 𝟏 𝟕 𝑥
𝒙 𝟏 𝟐
3 𝑥 5
𝒙 𝟓 𝟑
𝑥 3
𝒙 𝟑 𝟐
𝟖 𝒙
𝑥 1 8 𝒙
𝑥 1 2
𝑥 5 6
𝑥 7 2

26. Radical
Notation
Rational
Exponent
7 𝑥
𝒙 𝟏 𝟕 𝑥
𝒙 𝟏 𝟐
3 𝑥 5
𝒙 𝟓 𝟑
𝑥 3
𝒙 𝟑 𝟐
𝟖 𝒙
𝑥 1 8 𝒙
𝑥 1 2
𝟔 𝒙 𝟓
𝑥 5 6
𝑥 7 2

27. Radical
Notation
Rational
Exponent
7 𝑥
𝒙 𝟏 𝟕 𝑥
𝒙 𝟏 𝟐
3 𝑥 5
𝒙 𝟓 𝟑
𝑥 3
𝒙 𝟑 𝟐
𝟖 𝒙
𝑥 1 8 𝒙
𝑥 1 2
𝟔 𝒙 𝟓
𝑥 5 6
𝒙 𝟕
𝑥 7 2

28. Ex 4. Rewrite with a positive exponent. Simplify if possible

29. Ex 4. Rewrite with a positive exponent. Simplify if possible

30. Ex 4. Rewrite with a positive exponent. Simplify if possible

31. Ex 4. Rewrite with a positive exponent. Simplify if possible

32. Ex 4. Rewrite with a positive exponent. Simplify if possible

33. Ex 4. Rewrite with a positive exponent. Simplify if possible

34. Ex 4. Rewrite with a positive exponent. Simplify if possible

35. Note: The same rules that you learned before apply when working with rational exponents. 𝑏 𝑚 ⋅ 𝑏 𝑛 = 𝑏 𝑚+𝑛 𝑏 𝑚 𝑏 𝑛 = 𝑏 𝑚−𝑛 𝑏 𝑚 𝑛 = 𝑏 𝑚𝑛 𝑎𝑏 𝑛 = 𝑎 𝑛 𝑏 𝑛 𝑎 𝑏 𝑛 = 𝑎 𝑛 𝑏 𝑛 𝑏 −𝑛 = 1 𝑏 𝑛 𝑏 0 =1 When is an expression with rational exponents simplified? Basically, when you’re done using the above rules. That means…

36. 1. No parentheses around products or quotients
1. No parentheses around products or quotients. ex: 3𝑥 2 =𝟗 𝒙 𝟐 or 𝑥 3 2 = 𝒙 𝟐 𝟗 2. No powers raised to powers. ex: 𝑥 3 5 = 𝒙 𝟏𝟓 3. Each base occurs once. ex: 𝑥 2 𝑥 3 = 𝒙 𝟓 4. No negative or zero exponents. ex: 𝑥 −6 = 𝟏 𝒙 𝟔 and 𝑥 0 =𝟏 (Exception to “1”: if final answer is in radical notation, try to bring factors into a common radical. ex: 𝑎 1 2 𝑏 1 2 = 𝑎𝑏 1 2 = 𝑎𝑏 not 𝑎 1 2 𝑏 1 2 = 𝑎 𝑏 )

37. 1. No parentheses around products or quotients
1. No parentheses around products or quotients. ex: 3𝑥 2 =𝟗 𝒙 𝟐 or 𝑥 3 2 = 𝒙 𝟐 𝟗 2. No powers raised to powers. ex: 𝑥 3 5 = 𝒙 𝟏𝟓 3. Each base occurs once. ex: 𝑥 2 𝑥 3 = 𝒙 𝟓 4. No negative or zero exponents. ex: 𝑥 −6 = 𝟏 𝒙 𝟔 and 𝑥 0 =𝟏 (Exception to “1”: if final answer is in radical notation, try to bring factors into a common radical. ex: 𝑎 1 2 𝑏 1 2 = 𝑎𝑏 1 2 = 𝑎𝑏 not 𝑎 1 2 𝑏 1 2 = 𝑎 𝑏 )

38. 1. No parentheses around products or quotients
1. No parentheses around products or quotients. ex: 3𝑥 2 =𝟗 𝒙 𝟐 or 𝑥 3 2 = 𝒙 𝟐 𝟗 2. No powers raised to powers. ex: 𝑥 3 5 = 𝒙 𝟏𝟓 3. Each base occurs once. ex: 𝑥 2 𝑥 3 = 𝒙 𝟓 4. No negative or zero exponents. ex: 𝑥 −6 = 𝟏 𝒙 𝟔 and 𝑥 0 =𝟏 (Exception to “1”: if final answer is in radical notation, try to bring factors into a common radical. ex: 𝑎 1 2 𝑏 1 2 = 𝑎𝑏 1 2 = 𝑎𝑏 not 𝑎 1 2 𝑏 1 2 = 𝑎 𝑏 )

39. 1. No parentheses around products or quotients
1. No parentheses around products or quotients. ex: 3𝑥 2 =𝟗 𝒙 𝟐 or 𝑥 3 2 = 𝒙 𝟐 𝟗 2. No powers raised to powers. ex: 𝑥 3 5 = 𝒙 𝟏𝟓 3. Each base occurs once. ex: 𝑥 2 𝑥 3 = 𝒙 𝟓 4. No negative or zero exponents. ex: 𝑥 −6 = 𝟏 𝒙 𝟔 and 𝑥 0 =𝟏 (Exception to “1”: if final answer is in radical notation, try to bring factors into a common radical. ex: 𝑎 1 2 𝑏 1 2 = 𝑎𝑏 1 2 = 𝑎𝑏 not 𝑎 1 2 𝑏 1 2 = 𝑎 𝑏 )

40. 1. No parentheses around products or quotients
1. No parentheses around products or quotients. ex: 3𝑥 2 =𝟗 𝒙 𝟐 or 𝑥 3 2 = 𝒙 𝟐 𝟗 2. No powers raised to powers. ex: 𝑥 3 5 = 𝒙 𝟏𝟓 3. Each base occurs once. ex: 𝑥 2 𝑥 3 = 𝒙 𝟓 4. No negative or zero exponents. ex: 𝑥 −6 = 𝟏 𝒙 𝟔 and 𝑥 0 =𝟏 (Exception to “1”: if final answer is in radical notation, try to bring factors into a common radical. ex: 𝑎 1 2 𝑏 1 2 = 𝑎𝑏 1 2 = 𝑎𝑏 not 𝑎 1 2 𝑏 1 2 = 𝑎 𝑏 )

41. 1. No parentheses around products or quotients
1. No parentheses around products or quotients. ex: 3𝑥 2 =𝟗 𝒙 𝟐 or 𝑥 3 2 = 𝒙 𝟐 𝟗 2. No powers raised to powers. ex: 𝑥 3 5 = 𝒙 𝟏𝟓 3. Each base occurs once. ex: 𝑥 2 𝑥 3 = 𝒙 𝟓 4. No negative or zero exponents. ex: 𝑥 −6 = 𝟏 𝒙 𝟔 and 𝑥 0 =𝟏 (Exception to “1”: if final answer is in radical notation, try to bring factors into a common radical. ex: 𝑎 1 2 𝑏 1 2 = 𝑎𝑏 1 2 = 𝑎𝑏 not 𝑎 1 2 𝑏 1 2 = 𝑎 𝑏 )

42. 1. No parentheses around products or quotients
1. No parentheses around products or quotients. ex: 3𝑥 2 =𝟗 𝒙 𝟐 or 𝑥 3 2 = 𝒙 𝟐 𝟗 2. No powers raised to powers. ex: 𝑥 3 5 = 𝒙 𝟏𝟓 3. Each base occurs once. ex: 𝑥 2 𝑥 3 = 𝒙 𝟓 4. No negative or zero exponents. ex: 𝑥 −6 = 𝟏 𝒙 𝟔 and 𝑥 0 =𝟏 (Exception to “1”: if final answer is in radical notation, try to bring factors into a common radical. ex: 𝑎 1 2 𝑏 1 2 = 𝑎𝑏 1 2 = 𝑎𝑏 not 𝑎 1 2 𝑏 1 2 = 𝑎 𝑏 )

43. 1. No parentheses around products or quotients
1. No parentheses around products or quotients. ex: 3𝑥 2 =𝟗 𝒙 𝟐 or 𝑥 3 2 = 𝒙 𝟐 𝟗 2. No powers raised to powers. ex: 𝑥 3 5 = 𝒙 𝟏𝟓 3. Each base occurs once. ex: 𝑥 2 𝑥 3 = 𝒙 𝟓 4. No negative or zero exponents. ex: 𝑥 −6 = 𝟏 𝒙 𝟔 and 𝑥 0 =𝟏 (Exception to “1”: if final answer is in radical notation, try to bring factors into a common radical. ex: 𝑎 1 2 𝑏 1 2 = 𝑎𝑏 1 2 = 𝑎𝑏 not 𝑎 1 2 𝑏 1 2 = 𝑎 𝑏 )

44. Ex 5. Simplify (assume all variables represent positive numbers): ⋅ = 50 𝑥 𝑥 4 3 = 𝑥 − 3 5 𝑦 = 5 𝑦 2 10 𝑦 3 = 𝑥 ⋅ 3 𝑥 = 3 𝑥 = 6 𝑎 𝑏 2 ⋅ 3 𝑎 2 𝑏 =