272 CHAPTER 5 Number Theory and the Real Number System Exercises Warm Up Exercises In Exercises 1– 6, fill in the blanks with an appropriate word, phrase, or symbol(s). 1. With the product rule for exponents, the expression ⋅ x x 2 3 can be simplified to ________. x5 2. With the quotient rule for exponents, the expression x x 7 4 can be simplified to ________. x3 3. With the zero exponent rule, the expression 70 can be simplified to ________. 1 4. With the negative exponent rule, for ≠ x 0, the expression −x 5 can be simplified to ________. x 1 5 5. With the power rule for exponents, the expression x( )2 3 can be simplified to ________. x6 6. A number written in scientific notation is a number greater than or equal to 1 and less than 10 multiplied by a power of ________. Ten Practice the Skills In Exercises 7–22, evaluate the expression. Assume ≠ x 0. 7. a) 4 4 2 3 ⋅ 1024 b) ( 4) ( 4) 2 3 − ⋅ − −1024 8. a) 5 53 ⋅ 625 b) ( 5) ( 5)3 − ⋅ − 625 9. a) 3 3 9 7 9 b) − − ( 3) ( 3) 9 7 9 10. a) 4 4 5 2 64 b) − − ( 4) ( 4) 5 2 64 − 11. a) 60 1 b) 60 − −1 12. a) ( 6)0 − 1 b) ( 6)0 − − −1 13. a) x (6 )0 1 b) x6 0 6 14. a) x6 0 − −6 b) x ( 6 )0 − 1 15. a) −3 3 1 27 b) −7 2 1 49 16. a) −5 1 1 5 b) 2 4− 1 16 17. a) − −9 2 − 1 81 b) − − ( 9) 2 1 81 18. a) − −5 2 − 1 25 b) − − ( 5) 2 1 25 19. a) (2 )3 2 64 b) (3 )2 3 729 20. a) ⎛ ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 1 2 2 3 1 64 b) ⎛ − ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 1 2 2 3 1 64 21. a) ⋅ − 4 4 3 2 4 b) ⋅ − − 2 2 2 2 1 16 22. a) − − − − ( 5) ( 5) 2 2 1 625 b) − − − − ( 1) ( 1) 5 5 1 In Exercises 23–32, express the number in scientific notation. 23. 632,000 × 6.32 105 24. 95,000,000 × 9.5 107 25. 0.0032 × − 3.2 10 3 26. 0.0000263 × − 2.63 10 5 27. 0.56 × − 5.6 10 1 28. 0.00467 × − 4.67 10 3 29. 19,000 × 1.9 104 30. 1,260,000,000 × 1.26 109 31. 0.000186 1.86 10 4 × − 32. 0.0003 3 10 4 × − In Exercises 33– 40, express the number in decimal notation. 33. × 4.4 103 4400 34. × 8.2 106 8,200,000 35. 2.93 10 4 × − 0.000293 36. × − 7.19 10 5 0.0000719 37. × − 8.62 10 5 0.0000862 38. 2.19 10 4 × − 0.000219 39. × 2.01 100 2.01 40. × 4.6 101 46 In Exercises 41– 48, (a) perform the indicated operation without the use of a calculator and express each answer in decimal notation. (b) Confirm your answer from part (a) by using a scientific calculator to perform the operations. If the calculator displays the answer in scientific notation, convert the answer to decimal notation. 41. × × (7.2 10 ) (2 10 ) 2 3 1,440,000 42. × × (5.3 10 ) (3 10 ) 3 3 15,900,000 43. (5.1 10 )(3 10 ) 1 4 × × − 0.0153 44. × × − − (1.6 10 )(4 10 ) 2 3 0.000064 45. × × 7.5 10 3 10 6 4 250 46. × × − 2.7 10 9 10 3 2 30,000 47. × × − − 8.4 10 4 10 6 3 0.0021 48. × × 5.2 10 4 10 7 9 0.013 SECTION 5.6
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