Section Exercises

1. Is [latex]{2}^{3}[/latex] the same as [latex]{3}^{2}?[/latex] Explain.

2. When can you add two exponents?

3. What is the purpose of scientific notation?

4. Explain what a negative exponent does.

For the following exercises, simplify the given expression. Write answers with positive exponents.

5. [latex]{9}^{2}[/latex]

6. [latex]{15}^{-2}[/latex]

7. [latex]{3}^{2}\times {3}^{3}[/latex]

8. [latex]{4}^{4}\div 4[/latex]

9. [latex]{\left({2}^{2}\right)}^{-2}[/latex]

10. [latex]{\left(5 - 8\right)}^{0}[/latex]

11. [latex]{11}^{3}\div {11}^{4}[/latex]

12. [latex]{6}^{5}\times {6}^{-7}[/latex]

13. [latex]{\left({8}^{0}\right)}^{2}[/latex]

14. [latex]{5}^{-2}\div {5}^{2}[/latex]

For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents.

15. [latex]{4}^{2}\times {4}^{3}\div {4}^{-4}[/latex]

16. [latex]\frac{{6}^{12}}{{6}^{9}}[/latex]

17. [latex]{\left({12}^{3}\times 12\right)}^{10}[/latex]

18. [latex]{10}^{6}\div {\left({10}^{10}\right)}^{-2}[/latex]

19. [latex]{7}^{-6}\times {7}^{-3}[/latex]

20. [latex]{\left({3}^{3}\div {3}^{4}\right)}^{5}[/latex]

For the following exercises, express the decimal in scientific notation.

21. 0.0000314

22. 148,000,000

For the following exercises, convert each number in scientific notation to standard notation.

23. [latex]1.6\times {10}^{10}[/latex]

24. [latex]9.8\times {10}^{-9}[/latex]

For the following exercises, simplify the given expression. Write answers with positive exponents.

25. [latex]\frac{{a}^{3}{a}^{2}}{a}[/latex]

26. [latex]\frac{m{n}^{2}}{{m}^{-2}}[/latex]

27. [latex]{\left({b}^{3}{c}^{4}\right)}^{2}[/latex]

28. [latex]{\left(\frac{{x}^{-3}}{{y}^{2}}\right)}^{-5}[/latex]

29. [latex]a{b}^{2}\div {d}^{-3}[/latex]

30. [latex]{\left({w}^{0}{x}^{5}\right)}^{-1}[/latex]

31. [latex]\frac{{m}^{4}}{{n}^{0}}[/latex]

32. [latex]{y}^{-4}{\left({y}^{2}\right)}^{2}[/latex]

33. [latex]\frac{{p}^{-4}{q}^{2}}{{p}^{2}{q}^{-3}}[/latex]

34. [latex]{\left(l\times w\right)}^{2}[/latex]

35. [latex]{\left({y}^{7}\right)}^{3}\div {x}^{14}[/latex]

36. [latex]{\left(\frac{a}{{2}^{3}}\right)}^{2}[/latex]

37. [latex]{5}^{2}m\div {5}^{0}m[/latex]

38. [latex]\frac{{\left(16\sqrt{x}\right)}^{2}}{{y}^{-1}}[/latex]

39. [latex]\frac{{2}^{3}}{{\left(3a\right)}^{-2}}[/latex]

40. [latex]{\left(m{a}^{6}\right)}^{2}\frac{1}{{m}^{3}{a}^{2}}[/latex]

41. [latex]{\left({b}^{-3}c\right)}^{3}[/latex]

42. [latex]{\left({x}^{2}{y}^{13}\div {y}^{0}\right)}^{2}[/latex]

43. [latex]{\left(9{z}^{3}\right)}^{-2}y[/latex]

44. To reach escape velocity, a rocket must travel at the rate of [latex]2.2\times {10}^{6}[/latex] ft/min. Rewrite the rate in standard notation.

45. A dime is the thinnest coin in U.S. currency. A dime’s thickness measures [latex]2.2\times {10}^{6}[/latex] m. Rewrite the number in standard notation.

46. The average distance between Earth and the Sun is 92,960,000 mi. Rewrite the distance using scientific notation.

47. A terabyte is made of approximately 1,099,500,000,000 bytes. Rewrite in scientific notation.

48. The Gross Domestic Product (GDP) for the United States in the first quarter of 2014 was [latex]\$1.71496\times {10}^{13}[/latex]. Rewrite the GDP in standard notation.

49. One picometer is approximately [latex]3.397\times {10}^{-11}[/latex] in. Rewrite this length using standard notation.

50. The value of the services sector of the U.S. economy in the first quarter of 2012 was $10,633.6 billion. Rewrite this amount in scientific notation.

For the following exercises, use a graphing calculator to simplify. Round the answers to the nearest hundredth.

51. [latex]{\left(\frac{{12}^{3}{m}^{33}}{{4}^{-3}}\right)}^{2}[/latex]

52. [latex]{17}^{3}\div {15}^{2}{x}^{3}[/latex]

For the following exercises, simplify the given expression. Write answers with positive exponents.

53. [latex]{\left(\frac{{3}^{2}}{{a}^{3}}\right)}^{-2}{\left(\frac{{a}^{4}}{{2}^{2}}\right)}^{2}[/latex]

54. [latex]{\left({6}^{2}-24\right)}^{2}\div {\left(\frac{x}{y}\right)}^{-5}[/latex]

55. [latex]\frac{{m}^{2}{n}^{3}}{{a}^{2}{c}^{-3}}\cdot \frac{{a}^{-7}{n}^{-2}}{{m}^{2}{c}^{4}}[/latex]

56. [latex]{\left(\frac{{x}^{6}{y}^{3}}{{x}^{3}{y}^{-3}}\cdot \frac{{y}^{-7}}{{x}^{-3}}\right)}^{10}[/latex]

57. [latex]{\left(\frac{{\left(a{b}^{2}c\right)}^{-3}}{{b}^{-3}}\right)}^{2}[/latex]

58. Avogadro’s constant is used to calculate the number of particles in a mole. A mole is a basic unit in chemistry to measure the amount of a substance. The constant is [latex]6.0221413\times {10}^{23}[/latex]. Write Avogadro’s constant in standard notation.

59. Planck’s constant is an important unit of measure in quantum physics. It describes the relationship between energy and frequency. The constant is written as [latex]6.62606957\times {10}^{-34}[/latex]. Write Planck’s constant in standard notation.