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Understanding Exponents, Scientific Notation, and Sequences

This comprehensive resource covers the fundamental concepts of exponents, scientific notation, and geometric sequences. Learn how to simplify expressions involving exponents, convert numbers into scientific notation, and evaluate geometric sequences. The material includes examples, step-by-step solutions, and key properties of exponents. Whether you are preparing for exams or seeking a deeper understanding of mathematical principles, this guide is designed to help you master these essential topics.

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Understanding Exponents, Scientific Notation, and Sequences

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  1. Welcome To

  2. Scientific Notation Exponential Growth and Decay Properties of exponents Exponents Geometry Sequences $100 $100 $100 $100 $100 $200 $200 $200 $200 $200 $300 $300 $300 $300 $300 $400 $400 $400 $400 $400 $500 $500 $500 $500 $500

  3. Exponents for $100 Simplify: 4-3

  4. Answer 4-3 = 1/43 = 1/64 Back

  5. Exponentsfor $200 Simplify: (0.023454)0

  6. Answer (0.023454)0 = 1 Back

  7. Exponents for $300 Simplify: 3-2 * 40

  8. Answer 3-2 * 40 = 1/32 * 1 = 1/9 Back

  9. Exponents for $400 Simplify: 22/(3*2)-2

  10. Answer 22/(3*2)-2 = 4*(3*2)2 = 4* 62 = 4*36 = 144 Back

  11. Exponentsfor $500 Evaluate (2x3)/(3-2y-5) for x = -2 and y = 4

  12. Answer (2x3)/(3-2y-5) for x = -2 and y = 4 =2x3*32y5 = 2*9 *x3y5 = 18x3y5 = 18(-2)3(4)5 = 18*-8*1024 = -147456 Back

  13. Scientific Notationfor $100 Write the following number in scientific Notation: 3,450,000

  14. Answer 3,450,000 = 3.45 x 106 Back

  15. Scientific Notationfor $200 Write the following number in scientific Notation: .000073

  16. Answer .000073 = 7.3 x 10-5 Back

  17. Scientific Notationfor $300 Simplify. Write your answer in scientific notation: (3.24 x 10-4)(5.2 x 10-2)

  18. Answer (3.24 x 10-4)(5.2 x 10-2) = 3.24*5.2 x 10-4 * 10-2 = 16.848 x 10-6 = 1.6848 x 10-5 Back

  19. Scientific Notationfor $400 Simplify. Write your answer in scientific notation: (7.1 x 10-2)(2.3 x 104)

  20. Answer (7.1 x 10-2)(2.3 x 104) = 7.1*2.3 x 10-2 * 104 = 16.33 x 102 = 1.633 x 103 Back

  21. Scientific Notationfor $500 Simplify. Write your answer in scientific notation: ((1.3 x 103)(9.1 x 1012))2

  22. Answer ((1.3 x 103)(9.1 x 1012))2 (1.32 x 106)(9.12 x 1024) = (1.3*9.1)2 x 106 * 1024 = 139.9489 x 1030 = 1.399489 x 1032 Back

  23. Properties of Exponentsfor $100 Simplify: 2x-1 * 3x5

  24. Answer 2x-1 * 3x5 = 2*3*x-1+5 = 6x4 Back

  25. Properties of Exponentsfor $200 Simplify: (3x-3)/(9x4)

  26. Answer (3x-3)/(9x4) = (3/9)*x-3 – 4 = (1/3)x-7 = 1/(3x7) Back

  27. Properties of Exponentsfor $300 Simplify: (x-3x5)/(x2y0)

  28. Answer (x-3x5)/(x2y0) = (x-3+5)/(x2*1) = x2/x2 = 1 Back

  29. Properties of Exponentsfor $400 Simplify: (y-7y3)-1

  30. Answer (y-7y3)-1 = (y-7+3)-1 = (y-4)-1 = y4 Back

  31. Properties of Exponentsfor $500 Simplify: (9x-2y5)2 * (3y2x-1)-3

  32. Answer (9x-2y5)2 * (3y2x-1)-3 = (92x-2*2y5*2)2 * (3-3y2*-3x-1*-3)-3 = (81x-4y10) * ((1/27)y-6x3) = (81/27)(x-4x3)(y10y-6) = (3y4)/x Back

  33. Geometric Sequencesfor $100 Find the next three terms in the following geometric sequence: 3, 9, 27, 81, …

  34. Answer The common ratio is 3, so the next three terms are: 81*3 = 243 243*3 = 729 729*3 = 2187 243, 729, 2187 Back

  35. Geometric Sequencesfor $200 Write the formula for finding the nth term of a geometric sequence

  36. A(n) = a1 * (r)n-1 Where: n is the nth term a1 is the first term r is the common ratio Answer Back

  37. Geometric Sequencesfor $300 Find the 8th, 11th, and 13th terms in the following geometric sequence: A(n) = 4(3)n-1

  38. Answer A(n) = 4(3)n-1 A(8) = 4(3)8-1 = 4(3)7 = 8,748 A(11) = 4(3)11-1 = 4(3)10 = 236,196 A(13) = 4(3)13-1 = 4(3)12 = 2,125,764 Back

  39. Geometric Sequencesfor $400 Is the following sequence geometric, arithmetic, or neither? WHY? 4, 8, 12, 16, ….,

  40. Answer The sequence is arithmetic because the next term is found by adding 4 to the previous term. A geometric sequence would progress by multiplying, not adding. Back

  41. Geometric Sequencesfor $500 What is the difference between the equation for geometric sequences and the equation for exponential growth or decay?

  42. Answer The equation for a geometric sequence is to the power of n-1, while exponential growth and decay is to the power of x, not x-1. Geometric sequence: A(n) = a1 * (r)n-1 Exponential growth/decay: y = a * (b)x Back

  43. Exponential Growth and Decayfor $100 Is the following equation exponential growth or exponential decay? Why? A(n) = 5*(0.4)n

  44. Answer Exponential decay because r = 0.4 which means 0<r<1, thus telling you that it is exponential decay Back

  45. Exponential Growth and Decayfor $200 Is the following equation exponential growth or exponential decay? Why? A(n) = 0.2*(7)n

  46. Answer Exponential growth because r = 7 which means r>1, thus telling you that it is exponential growth Back

  47. Exponential Growth and Decayfor $300 Write an equation to model the following situation (in other words, write an equation to find the number of balls left after n days) Josh started with 200 balls, but loses half of them every day

  48. Answer A(n) = 200(0.5)n Back

  49. Exponential Growth and Decayfor $400 Joe invested $3200 in an account that earned 4% interest compounded every 3 months. Find the account balance after 5 years.

  50. Answer y = a*bx where y = final balance, a = initial amount, b = interest rate and x = the number of times the interest is compounded. So, a = 3200, b = 4% = 1 + .04 = 1.04 and x = 20 (4 per year for 5 years) y = 3200(1.04)20 = $7011.59 Back

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