Mat 092 section 12.1 the power and product rules for exponents

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1. Slide - 1Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G 2 Exponents and Polynomials 12 2. Slide - 2Copyright © 2018, 2014, 2010…
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  • 1. Slide - 1Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G 2 Exponents and Polynomials 12
  • 2. Slide - 2Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G 1. Use exponents. 2. Use the product rule for exponents. 3. Use the rule (am)n = amn. 4. Use the rule (ab)m = ambm. 5. Use the rule (a/b)m = am/bm. 6. Use combinations of the rules for exponents. 7. Use the rules for exponents in a geometry application. Objectives 12.1 The Product Rule and Power Rules for Exponents
  • 3. Slide - 3Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Use Exponents Exponent (or Power) Base 6 factors of 2 2 = 2 · 2 · 2 · 2 · 2 · 2 6 The exponential expression is 26, read “2 to the sixth power” or simply “2 to the sixth.” Example Write 2 · 2 · 2 · 2 · 2 · 2 in exponential form and evaluate. Since 2 occurs as a factor 6 times, the base is 2 and the exponent is 6. = 64
  • 4. Slide - 4Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Evaluating Exponential Expressions (a) 2 = 2 · 2 · 2 · 24 = 16 Base Exponent 2 4 (b) – 2 4 = –16 2 4 4= –1 · 2 = –1 · 2 · 2 · 2 · 2 (c) (–2)4 = 16 –2 4 = (–2) (–2) (–2) (–2) Example Evaluate. Name the base and the exponent.
  • 5. Slide - 5Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Expression Base Exponent Example Use Exponents CAUTION – an (–a) n In summary, and are not necessarily the same.– a n (–a) n a n (–a) n – 52 (– 5)2 = – ( 5 · 5 ) = – 25 = (–5) (–5) = 25
  • 6. Slide - 6Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Use the Product Rule for Exponents Product Rule for Exponents For any positive integers m and n, a m · a n = a m + n (Keep the same base and add the exponents.) 53 · 54 Example: ( 5 · 5 · 5 ) = 5 3 + 4 ( 5 · 5 · 5 · 5 ) = 57
  • 7. Slide - 7Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Use the Product Rule for Exponents CAUTION Do not multiply the bases when using the product rule. Keep the same base and add the exponents. Example: 3 4 7 7 5 5 5 not 25 . 
  • 8. Slide - 8Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Using the Product Rule = 75 + 85 8(a) 7 · 7 = 713 = y1 + 55(c) y · y = y65= y · y1 = (– 2)4 + 54 5(b) (–2) (–2) = (– 2)9 Example Use the product rule for exponents to simplify, if possible.
  • 9. Slide - 9Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Using the Product Rule = n 2 + 42 4(d) n · n = n 6 The product rule does not apply because the bases are different. 2 2 (e) 3 · 2 The product rule does not apply because it is a sum, not a product. 3 2 (f) 2 + 2 = 8 + 4 = 12 = 9 · 4 = 36 Example (cont) Use the product rule for exponents to simplify, if possible.
  • 10. Slide - 10Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Using the Product Rule Add the exponents. 3 6 Multiply 5 4 .y y Example 3 6 5 4y y   3 6 5 4 y y   3 6 20y   Multiply; product rule Commutative and associative properties 9 20y
  • 11. Slide - 11Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Use the Product Rule for Exponents CAUTION Be sure you understand the difference between adding and multiplying exponential expressions. For example, 2 2 7k + 3k 22 = ( 7 + 3 )k = 10k , 42 + 2 = ( 7 · 3 )k = 21k . 2 2 7k 3kbut, Add. Multiply.
  • 12. Slide - 12Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Use the Rule (am)n = amn Power Rule (a) for Exponents For any positive integers m and n, m n ( a ) Example: 3 2 ( 4 ) 3 · 2 = 4 (Raise a power to a power by multiplying exponents.) 6 = 4 . m n = a .
  • 13. Slide - 13Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Use the Rule (am)n = amn Example Use power rule (a) for exponents to simplify. (a) ( 3 )2 5 = 32 · 5 = 3 10 (b) ( 4 )8 6 = 48 · 6 = 4 48 (c) ( n )7 3 = n 7 · 3 = n 21 (d) ( 2 )5 8 = 2 5 · 8 = 2 40
  • 14. Slide - 14Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Use the Rule (ab)m = ambm Power Rule (b) for Exponents For any positive integer m, m ( ab ) m m = a b . Example: 3 ( 5h ) (Raise a product to a power by raising each factor to the power.) 3 = 5 h .3
  • 15. Slide - 15Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Use the Rule (ab)m = ambm Example Use power rule (b) for exponents to simplify. (a) ( 2abc ) 4 = 2 a b c4 4 4 4 = 16 a b c4 4 4 Power rule (b) = 5 ( x y )2 6 Power rule (b)(b) 5 ( x y ) 23 = 5x y2 6
  • 16. Slide - 16Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Use the Rule (ab)m = ambm Example (cont) Use power rule (b) for exponents to simplify. Power rule (b) (c) 7 ( 2m n p )75 3 = 7 [ 2 ( m ) ( n ) ( p ) ]3 5 3 71 3 3 Power rule (a)= 7 [ 8 m n p ]3 15 21 = 56 m n p3 15 21
  • 17. Slide - 17Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Use the Rule (ab)m = ambm Example (cont) Use power rule (b) for exponents to simplify. Power rule (b) (d) ( –3 ) 54 Power rule (a) = ( – 1 · 3 ) 54 = ( – 1 ) ( 3 ) 545 = – 1 · 3 20 = – 3 20 – a = – 1 · a
  • 18. Slide - 18Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Use the Rule (ab)m = ambm CAUTION Power rule (b) does not apply to a sum:     2 22 2 2 2 5 5 , but 5 5 .y y y y  
  • 19. Slide - 19Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Use the Rule (a/b)m = am/bm Power Rule (c) for Exponents For any positive integer m, (Raise a quotient to a power by raising both the numerator and the denominator to the power.) ma b ma bm = Example: 23 4 23 4 2 = ( b ≠ 0 ).
  • 20. Slide - 20Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Use the Rule (a/b)m = am/bm Example Use power rule (c) for exponents to simplify. 42 5 42 5 4 =(a) 16 625 = 7x y 7x y 7 =(b) y ≠ 0
  • 21. Slide - 21Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Rules for Exponents For any positive integers m and n: Examples Product rule am · an = am + n 34 · 35 = 39 Power rules m n ( a ) m n = a(a) 4 5 ( 2 ) 20 = 2 m m m (b) ( ab ) = a b 3 ( 4k ) 3 = 4 k 3 (c) ma b ma bm = ( b ≠ 0 ). 34 7 = 34 7 3
  • 22. Slide - 22Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Use Combinations of the Rules for Exponents Example Simplify each expression. Power rule (c) Multiply fractions. (a) · 35 23 4 = · 23 42 53 1 = 23 · 4 ·2 53 1 Product rule= 3 42 7
  • 23. Slide - 23Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Use Combinations of the Rules for Exponents Example (cont) Simplify each expression. Product rule (b) ( 7m n ) ( 7m n )22 3 5 = ( 7m n )2 8 Power rule (b)= 7 m n16 88
  • 24. Slide - 24Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Use Combinations of the Rules for Exponents Example (cont) Simplify each expression. (c) ( 2x y ) ( 2 x y ) 1063 7 4 = ( 2 ) ( x ) ( y ) · ( 2 ) ( x ) ( y )10 7 46 3 6 6 10 10 = 2 · x · y · 2 · x · y10 70 406 18 6 Power rule (a) = 2 · 2 · x · x · y · y70 6 406 10 18 Commutative and associative properties = 2 · x · y16 88 46 Product rule Power rule (b)
  • 25. Slide - 25Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Use Combinations of the Rules for Exponents Example (cont) Simplify each expression. Power rule (b) (d) ( – g h ) ( – g h )3 2 5 32 = ( – 1 g h ) ( – 1 g h )3 2 5 32 = ( – 1 ) ( g ) ( h ) ( – 1 ) ( g ) ( h )2 26 3 6 15 Product rule= ( – 1 ) ( g ) ( h )5 12 17 = – 1 g h12 17
  • 26. Slide - 26Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Use the Rules for Exponents in a Geometry Application Example Find a polynomial that represents the area of the geometric figure. (a) Use the formula for the area of a rectangle, A = LW. A = ( 4x )( 2x )3 2 A = 8x 5 Product rule 4x 3 2x 2
  • 27. Slide - 27Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G Use the Rules for Exponents in a Geometry Application Example Find a polynomial that represents the area of the geometric figure. A = 12n9 Product rule 8n5 3n4 A = ( 3n ) ( 8n )4 51 2 (b) Use the formula for the area of a triangle, A = LW.1 2 A = ( 24n )91 2
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