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- 1. Review on Exponents
- 2. Let’s review the basics of exponential notation. Review on Exponents
- 3. base exponent The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN Let’s review the basics of exponential notation. Review on Exponents N times
- 4. base exponent Rules of Exponents The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN Let’s review the basics of exponential notation. Review on Exponents N times
- 5. base exponent Multiply–Add Rule: Rules of Exponents Divide–Subtract Rule: The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN Power–Multiply Rule: Let’s review the basics of exponential notation. Review on Exponents N times
- 6. base exponent Multiply–Add Rule: ANAK = AN+K Rules of Exponents Divide–Subtract Rule: The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN Power–Multiply Rule: Let’s review the basics of exponential notation. Review on Exponents N times
- 7. base exponent Multiply–Add Rule: ANAK = AN+K Rules of Exponents Divide–Subtract Rule: AN AK = AN – K The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN Power–Multiply Rule: Let’s review the basics of exponential notation. Review on Exponents N times
- 8. base exponent Multiply–Add Rule: ANAK = AN+K Rules of Exponents Divide–Subtract Rule: AN AK = AN – K The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN Power–Multiply Rule: (AN)K = ANK Let’s review the basics of exponential notation. Review on Exponents N times
- 9. base exponent Multiply–Add Rule: ANAK = AN+K Rules of Exponents Divide–Subtract Rule: AN AK = AN – K The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN Power–Multiply Rule: (AN)K = ANK Let’s review the basics of exponential notation. For example, x9x5 =x14 , x9 x5 = x9–5 = x4, and (x9)5 = x45. Review on Exponents N times
- 10. base exponent Multiply–Add Rule: ANAK = AN+K Rules of Exponents Divide–Subtract Rule: AN AK = AN – K The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN Power–Multiply Rule: (AN)K = ANK Let’s review the basics of exponential notation. For example, x9x5 =x14 , x9 x5 = x9–5 = x4, and (x9)5 = x45. Review on Exponents N times These particular operation–conversion rules appear often in other forms in mathematics.
- 11. base exponent Multiply–Add Rule: ANAK = AN+K Rules of Exponents Divide–Subtract Rule: AN AK = AN – K The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN Power–Multiply Rule: (AN)K = ANK Let’s review the basics of exponential notation. For example, x9x5 =x14 , x9 x5 = x9–5 = x4, and (x9)5 = x45. Review on Exponents N times These particular operation–conversion rules appear often in other forms in mathematics. Hence their names, the Multiply– Add Rule, the Divide–Subtract Rule, the Power–Multiply Rule, are important.
- 12. base exponent Multiply–Add Rule: ANAK = AN+K Rules of Exponents Divide–Subtract Rule: AN AK = AN – K The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN Power–Multiply Rule: (AN)K = ANK Let’s review the basics of exponential notation. For example, x9x5 =x14 , x9 x5 = x9–5 = x4, and (x9)5 = x45. Review on Exponents N times These particular operation–conversion rules appear often in other forms in mathematics. Hence their names, the Multiply– Add Rule, the Divide–Subtract Rule, the Power–Multiply Rule, are important. Let’s extend the definition to negative and fractional exponents.
- 13. Since = 1 A1 A1 The Exponential Functions
- 14. Since = 1 = A1 – 1 = A0A1 A1 The Exponential Functions
- 15. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 The Exponential Functions
- 16. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 0-Power Rule: A0 = 1 The Exponential Functions
- 17. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 0-Power Rule: A0 = 1 Since = 1 AK A0 AK The Exponential Functions
- 18. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, 1 AK A0 AK The Exponential Functions
- 19. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the Negative Power Rule. 1 AK A0 AK Negative Power Rule: A–K = 1 AK The Exponential Functions
- 20. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the Negative Power Rule. 1 AK A0 AK Negative Power Rule: A–K = 1 AK Since (A )k = A = (A1/k )k, k The Exponential Functions
- 21. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the Negative Power Rule. 1 AK A0 AK Negative Power Rule: A–K = 1 AK Since (A )k = A = (A1/k )k, hence A1/k = A. k k The Exponential Functions
- 22. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the Negative Power Rule. 1 AK A0 AK Negative Power Rule: A–K = 1 AK Fractional Powers: A1/k = A. k The Exponential Functions Since (A )k = A = (A1/k )k, hence A1/k = A. k k
- 23. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the Negative Power Rule. 1 AK A0 AK Negative Power Rule: A–K = 1 AK Fractional Powers: A1/k = A. k The Exponential Functions For a general fractional exponent, we interpret the operations step by step by doing the numerator of the exponent last. Since (A )k = A = (A1/k )k, hence A1/k = A. k k
- 24. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the Negative Power Rule. 1 AK A0 AK Negative Power Rule: A–K = 1 AK Fractional Powers: A1/k = A. k The Exponential Functions Since (A )k = A = (A1/k )k, hence A1/k = A. k k For a general fractional exponent, we interpret the operations step by step by doing the numerator of the exponent last.
- 25. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the Negative Power Rule. 1 AK A0 AK Negative Power Rule: A–K = 1 AK Example A. Simplify. b. 91/2 = a. Fractional Powers: A1/k = A. k The Exponential Functions For a general fractional exponent, we interpret the operations step by step by doing the numerator of the exponent last. 9–2 = c. 9 –3/2 = Since (A )k = A = (A1/k )k, hence A1/k = A. k k
- 26. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the Negative Power Rule. 1 AK A0 AK Negative Power Rule: A–K = 1 AK Example A. Simplify. 1 92 1 81 b. 91/2 = a. Fractional Powers: A1/k = A. k The Exponential Functions For a general fractional exponent, we interpret the operations step by step by doing the numerator of the exponent last. 9–2 = = c. 9 –3/2 = Since (A )k = A = (A1/k )k, hence A1/k = A. k k
- 27. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the Negative Power Rule. 1 AK A0 AK Negative Power Rule: A–K = 1 AK Example A. Simplify. 1 92 1 81 b. 91/2 = √9 = 3 a. Fractional Powers: A1/k = A. k The Exponential Functions For a general fractional exponent, we interpret the operations step by step by doing the numerator of the exponent last. 9–2 = = c. 9 –3/2 = Since (A )k = A = (A1/k )k, hence A1/k = A. k k
- 28. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the Negative Power Rule. 1 AK A0 AK Negative Power Rule: A–K = 1 AK Example A. Simplify. 1 92 1 81 b. 91/2 = √9 = 3 a. Fractional Powers: A1/k = A. k The Exponential Functions For a general fractional exponent, we interpret the operations step by step by doing the numerator of the exponent last. 9–2 = = c. 9 –3/2 = (9½)–3 Since (A )k = A = (A1/k )k, hence A1/k = A. k k Pull the numerator outside to take the root and simplify the base first.
- 29. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the Negative Power Rule. 1 AK A0 AK Negative Power Rule: A–K = 1 AK Example A. Simplify. 1 92 1 81 b. 91/2 = √9 = 3 a. Fractional Powers: A1/k = A. k The Exponential Functions For a general fractional exponent, we interpret the operations step by step by doing the numerator of the exponent last. 9–2 = = c. 9 –3/2 = (9½)–3 = 3–3 Since (A )k = A = (A1/k )k, hence A1/k = A. k k Pull the numerator outside to take the root and simplify the base first.
- 30. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the Negative Power Rule. 1 AK A0 AK Negative Power Rule: A–K = 1 AK Example A. Simplify. 1 92 1 81 1 33 b. 91/2 = √9 = 3 a. Fractional Powers: A1/k = A. k The Exponential Functions For a general fractional exponent, we interpret the operations step by step by doing the numerator of the exponent last. 9–2 = = c. 9 –3/2 = (9½)–3 = 3–3 = = 1 27 Since (A )k = A = (A1/k )k, hence A1/k = A. k k Pull the numerator outside to take the root and simplify the base first.
- 31. Power Equations and Calculator Inputs
- 32. Power Equations and Calculator Inputs Power Equations The solution to the equation x 3 = –8 is
- 33. Power Equations and Calculator Inputs Power Equations The solution to the equation x 3 = –8 is x = √–8 = –2. 3
- 34. Power Equations and Calculator Inputs Power Equations The solution to the equation x 3 = –8 is x = √–8 = –2. 3 Using fractional exponent notation, we write these steps as if x3 = –8 then
- 35. Power Equations and Calculator Inputs Power Equations The solution to the equation x 3 = –8 is x = √–8 = –2. 3 Using fractional exponent notation, we write these steps as if x3 = –8 then x = (–8)1/3 The reciprocal of the power 3
- 36. Power Equations and Calculator Inputs Power Equations The solution to the equation x 3 = –8 is x = √–8 = –2. 3 Using fractional exponent notation, we write these steps as if x3 = –8 then x = (–8)1/3 = –2. The reciprocal of the power 3
- 37. Power Equations and Calculator Inputs Power Equations The solution to the equation x 3 = –8 is x = √–8 = –2. 3 Using fractional exponent notation, we write these steps as if x3 = –8 then x = (–8)1/3 = –2. (Rational) Power equations are equations of the type xP/Q = c. The reciprocal of the power 3
- 38. Power Equations and Calculator Inputs Power Equations The solution to the equation x 3 = –8 is x = √–8 = –2. 3 Using fractional exponent notation, we write these steps as if x3 = –8 then x = (–8)1/3 = –2. (Rational) Power equations are equations of the type xP/Q = c. To solve them, we take the reciprocal power, that is, if xP/Q = c, The reciprocal of the power 3
- 39. Power Equations and Calculator Inputs Power Equations The solution to the equation x 3 = –8 is x = √–8 = –2. 3 Using fractional exponent notation, we write these steps as if x3 = –8 then x = (–8)1/3 = –2. (Rational) Power equations are equations of the type xP/Q = c. To solve them, we take the reciprocal power, that is, if xP/Q = c, then x = (±) c Q/P. The reciprocal of the power 3 The reciprocal of the power P/Q
- 40. Power Equations and Calculator Inputs Power Equations The solution to the equation x 3 = –8 is x = √–8 = –2. 3 Using fractional exponent notation, we write these steps as if x3 = –8 then x = (–8)1/3 = –2. (Rational) Power equations are equations of the type xP/Q = c. To solve them, we take the reciprocal power, that is, if xP/Q = c, then x = (±) c Q/P. Note that xP/Q may not exist, or that sometime we get both (±) xP/Q solutions means that sometimes. The reciprocal of the power P/Q The reciprocal of the power 3
- 41. Power Equations and Calculator Inputs Example A. Solve for the real solutions. a. x3 = 64 b. x2 = 64 c. x2 = –64 d. x –2/3 = 64
- 42. Power Equations and Calculator Inputs Example A. Solve for the real solutions. a. x3 = 64 x = 641/3 b. x2 = 64 c. x2 = –64 d. x –2/3 = 64
- 43. Power Equations and Calculator Inputs Example A. Solve for the real solutions. a. x3 = 64 x = 641/3 or that 3 x = √64 = 4. b. x2 = 64 c. x2 = –64 d. x –2/3 = 64
- 44. Power Equations and Calculator Inputs Example A. Solve for the real solutions. a. x3 = 64 x = 641/3 or that 3 We note that this is the only solution. x = √64 = 4. b. x2 = 64 c. x2 = –64 d. x –2/3 = 64
- 45. Power Equations and Calculator Inputs Example A. Solve for the real solutions. a. x3 = 64 x = 641/3 or that 3 We note that this is the only solution. x = √64 = 4. b. x2 = 64 x = 641/2 or that x = √64 = 8. c. x2 = –64 d. x –2/3 = 64
- 46. Power Equations and Calculator Inputs Example A. Solve for the real solutions. a. x3 = 64 x = 641/3 or that 3 We note that this is the only solution. x = √64 = 4. b. x2 = 64 x = 641/2 or that We note that both ±8 are solutions. x = √64 = 8. c. x2 = –64 d. x –2/3 = 64
- 47. Power Equations and Calculator Inputs Example A. Solve for the real solutions. a. x3 = 64 x = 641/3 or that 3 We note that this is the only solution. x = √64 = 4. b. x2 = 64 x = 641/2 or that We note that both ±8 are solutions. x = √64 = 8. c. x2 = –64 x = (–64)1/2 which is UDF. d. x –2/3 = 64
- 48. Power Equations and Calculator Inputs Example A. Solve for the real solutions. a. x3 = 64 x = 641/3 or that 3 We note that this is the only solution. x = √64 = 4. b. x2 = 64 x = 641/2 or that We note that both ±8 are solutions. x = √64 = 8. c. x2 = –64 x = (–64)1/2 which is UDF. (In fact what most calculators return as the answer meaning that there is no real solutions.) d. x –2/3 = 64
- 49. Power Equations and Calculator Inputs Example A. Solve for the real solutions. a. x3 = 64 x = 641/3 or that 3 We note that this is the only solution. x = √64 = 4. b. x2 = 64 x = 641/2 or that We note that both ±8 are solutions. x = √64 = 8. c. x2 = –64 x = (–64)1/2 which is UDF. (In fact what most calculators return as the answer meaning that there is no real solutions.) d. x –2/3 = 64 x = 64–3/2
- 50. Power Equations and Calculator Inputs Example A. Solve for the real solutions. a. x3 = 64 x = 641/3 or that 3 We note that this is the only solution. x = √64 = 4. b. x2 = 64 x = 641/2 or that We note that both ±8 are solutions. x = √64 = 8. c. x2 = –64 x = (–64)1/2 which is UDF. (In fact what most calculators return as the answer meaning that there is no real solutions.) d. x –2/3 = 64 x = 64–3/2 x = (√64)–3
- 51. Power Equations and Calculator Inputs Example A. Solve for the real solutions. a. x3 = 64 x = 641/3 or that 3 We note that this is the only solution. x = √64 = 4. b. x2 = 64 x = 641/2 or that We note that both ±8 are solutions. x = √64 = 8. c. x2 = –64 x = (–64)1/2 which is UDF. (In fact what most calculators return as the answer meaning that there is no real solutions.) d. x –2/3 = 64 x = 64–3/2 x = (√64)–3 = 8–3 = 1/512.
- 52. Power Equations and Calculator Inputs Finally, for linear form of the power equations, we solve for the power term first , then apply the reciprocal power to find x. e. 2x2/3 – 7 = 1
- 53. Power Equations and Calculator Inputs Finally, for linear form of the power equations, we solve for the power term first , then apply the reciprocal power to find x. e. 2x2/3 – 7 = 1 2x2/3 = 8
- 54. Power Equations and Calculator Inputs Finally, for linear form of the power equations, we solve for the power term first , then apply the reciprocal power to find x. e. 2x2/3 – 7 = 1 2x2/3 = 8 x2/3 = 4 x = 43/2 x = (√4)3 = 8. We note that both ±8 are solutions. Mathematics Inputs in Text Format Most digital calculation devices such as calculators, smart phone apps or computer software accept inputs in the text format. Besides the “+” , “–”, for addition and subtraction we use “ * ” for multiplication, and “/” for the division operation. The power operation is represented by “^”. For example, the fraction is inputted as “3/4”, and the quantity 34 is “3^4”. All executions of such inputs follow the order of operations. 3 4
- 55. Power Equations and Calculator Inputs Finally, for linear form of the power equations, we solve for the power term first , then apply the reciprocal power to find x. e. 2x2/3 – 7 = 1 2x2/3 = 8 x2/3 = 4
- 56. Power Equations and Calculator Inputs Finally, for linear form of the power equations, we solve for the power term first , then apply the reciprocal power to find x. e. 2x2/3 – 7 = 1 2x2/3 = 8 x2/3 = 4 x = 43/2
- 57. Power Equations and Calculator Inputs Finally, for linear form of the power equations, we solve for the power term first , then apply the reciprocal power to find x. e. 2x2/3 – 7 = 1 2x2/3 = 8 x2/3 = 4 x = 43/2 x = (√4)3 = 8. We note that both ±8 are solutions.
- 58. Power Equations and Calculator Inputs Finally, for linear form of the power equations, we solve for the power term first , then apply the reciprocal power to find x. e. 2x2/3 – 7 = 1 2x2/3 = 8 x2/3 = 4 x = 43/2 x = (√4)3 = 8. We note that both ±8 are solutions. Mathematics Inputs in Text Format Most digital calculation devices such as calculators, smart phone apps or computer software accept inputs in the text format.
- 59. Power Equations and Calculator Inputs Finally, for linear form of the power equations, we solve for the power term first , then apply the reciprocal power to find x. e. 2x2/3 – 7 = 1 2x2/3 = 8 x2/3 = 4 x = 43/2 x = (√4)3 = 8. We note that both ±8 are solutions. Mathematics Inputs in Text Format Most digital calculation devices such as calculators, smart phone apps or computer software accept inputs in the text format. Besides the “+” , “–”, for addition and subtraction we use “ * ” for multiplication, and “/” for the division operation. The power operation is represented by “^”.
- 60. Power Equations and Calculator Inputs Finally, for linear form of the power equations, we solve for the power term first , then apply the reciprocal power to find x. e. 2x2/3 – 7 = 1 2x2/3 = 8 x2/3 = 4 x = 43/2 x = (√4)3 = 8. We note that both ±8 are solutions. Mathematics Inputs in Text Format Most digital calculation devices such as calculators, smart phone apps or computer software accept inputs in the text format. Besides the “+” , “–”, for addition and subtraction we use “ * ” for multiplication, and “/” for the division operation. The power operation is represented by “^”. For example, the fraction is inputted as “3/4”, and the quantity 34 is “3^4”. All executions of such inputs follow the order of operations. 3 4
- 61. Power Equations and Calculator Inputs Example B. Input and execute on a graphing calculator or software. Many common input mistakes happen for expressions involving division or taking powers. 3 2 4 2 + 6
- 62. Power Equations and Calculator Inputs Example B. Input and execute on a graphing calculator or software. Many common input mistakes happen for expressions involving division or taking powers. 3 2 4 2 + 6 The correct text input is (2+6)/(4^(3/2)) to get the correct answer of 1.
- 63. Power Equations and Calculator Inputs Example B. Input and execute on a graphing calculator or software. Many common input mistakes happen for expressions involving division or taking powers. 3 2 4 2 + 6 The correct text input is (2+6)/(4^(3/2)) to get the correct answer of 1. In general, when in doubt, insert ( )’s in the input to clarify the order of operations.

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