Introduction to Real Numbers

For the following exercises, simplify the given expression.

1. [latex]6 \div 2 − ( 81 \div 3^2 )[/latex]
2. [latex]-2 \times \left[16 \div (8 - 4)^2 \right]^2[/latex]
3. [latex]3(5-8)[/latex]
4. [latex]12 \div (36 \div 9) + 6[/latex]
5. [latex]3 − 12 \div 2 + 19[/latex]
6. [latex]5+(6+4)−11[/latex]
7. [latex]14 \times 3 \div 7 − 6[/latex]
8. [latex]6+2 \times 2-1[/latex]
9. [latex]9 + 4 (2^2)[/latex]
10. [latex]25 \div 5^2 − 7[/latex]
11. [latex]2 \times 4-9(-1)[/latex]
12. [latex]12(3-1) \div 6[/latex]

For the following exercises, evaluate the expression using the given value of the variable.

13. [latex]4y + 8 − 2y \text{ for } y = 3[/latex]
14. [latex]4z − 2z(1 + 4) − 36 \text{ for } z = 5[/latex]
15. [latex]-(2x)^2 + 1 + 3 \text{ for } x = 2[/latex]
16. [latex]2(11c − 4) − 36 \text{ for } c = 0[/latex]
17. [latex]\frac{1}{4} (8w − 4^2) \text{ for } w = 1[/latex]

For the following exercises, simplify the expression.

18. [latex]2y − (4)^2y − 11[/latex]
19. [latex]8b − 4b(3) + 1[/latex]
20. [latex]7z − 3 + z \times 6^2[/latex]
21. [latex]9 (y + 8) − 27[/latex]
22. [latex]6+12b-3 \times 6b[/latex]
23. [latex](\frac{4}{9})^2 \times 27x[/latex]
24. [latex]9x + 4x (2 + 3) − 4(2x + 3x)[/latex]

For the following exercises, consider this scenario: Fred earns [latex]$40[/latex] at the community garden. He spends [latex]$10[/latex] on a streaming subscription, puts half of what is left in a savings account, and gets another [latex]$5[/latex] for walking his neighbor’s dog.

25. Write the expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations.
26. How much money does Fred keep?

For the following exercises, solve the given problem.

27. According to the U.S. Mint, the diameter of a quarter is [latex]0.955[/latex] inches. The circumference of the quarter would be the diameter multiplied by [latex]\pi[/latex]. Is the circumference of a quarter a whole number, a rational number, or an irrational number?
28. Jessica and her roommate, Adriana, have decided to share a change jar for joint expenses. Jessica put her loose change in the jar first, and then Adriana put her change in the jar. We know that it does not matter in which order the change was added to the jar. What property of addition describes this fact?

For the following exercises, consider this scenario: There is a mound of [latex]g[/latex] pounds of gravel in a quarry. Throughout the day, [latex]400[/latex] pounds of gravel is added to the mound. Two orders of [latex]600[/latex] pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has [latex]1,200[/latex] pounds of gravel.

29. Write the equation that describes the situation.
30. Solve for [latex]g[/latex].

For the following exercise, solve the given problem.

31. Ramon runs the marketing department at their company. Their department gets a budget every year, and every year, they must spend the entire budget without going over. If they spend less than the budget, then the department gets a smaller budget the following year. At the beginning of this year, Ramon got [latex]$2.5[/latex] million for the annual marketing budget. They must spend the budget such that [latex]2,500,000 - x = 0[/latex]. What property of addition tells us what the value of [latex]x[/latex] must be?

Exponents and Scientific Notation

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

32. [latex]9^2[/latex]
33. [latex]3^2 \times 3^3[/latex]
34. [latex](2^2)^{-2}[/latex]
35. [latex]11^3 \div 11^4[/latex]
36. [latex](8^0)^2[/latex]

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

37. [latex]4^2 \times 4^3 \div 4^{-4}[/latex]
38. [latex](12^3 \times 12)^{10}[/latex]
39. [latex]7^{-6} \times 7^{-3}[/latex]

Express the decimal in scientific notation.

40. [latex]0.0000314[/latex]

Convert the number in scientific notation to standard notation.

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

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

42. [latex]\frac{a^3 a^2}{a}[/latex]
43. [latex](b^3 c^4 )^2[/latex]
44. [latex]ab^2 \div d^{-3}[/latex]
45. [latex]\frac{m^4}{n^0}[/latex]
46. [latex]\frac{p^{-4}q^2}{p^2q^{-3}}[/latex]
47. [latex](y^7)^3 \div x^{14}[/latex]
48. [latex](25m) \div (5^{0}m)[/latex]
49. [latex]\frac{2^3}{(3a)^{-2}}[/latex]
50. [latex](b^{-3}c)^3[/latex]
51. [latex](9z^3)^{-2}y[/latex]

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

52. A dime is the thinnest coin in U.S. currency. A dime’s thickness measures [latex]1.35 \times 10^{−3}[/latex] m. Rewrite the number in standard notation.

53. A terabyte is made of approximately [latex]1,099,500,000,000[/latex] bytes. Rewrite in scientific notation.

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

Roots and Rational Exponents

For the following exercises, simplify each expression.

55. [latex]\sqrt{256}[/latex]
56. [latex]\sqrt{4(9+16)}[/latex]
57. [latex]\sqrt{196}[/latex]
58. [latex]\sqrt{98}[/latex]
59. [latex]\sqrt{\frac{81}{5}}[/latex]
60. [latex]\sqrt{169} + \sqrt{144}[/latex]
61. [latex]\frac{18}{\sqrt{162}}[/latex]
62. [latex]14\sqrt{6} - 6 \sqrt{24}[/latex]
63. [latex]\sqrt{150}[/latex]
64. [latex](\sqrt{42})(\sqrt{30})[/latex]
65. [latex]\sqrt{\frac{4}{225}}[/latex]
66. [latex]\sqrt{\frac{360}{361}}[/latex]
67. [latex]\frac{8}{1 - \sqrt{17}}[/latex]
68. [latex]\sqrt[3]{128} + 3 \sqrt[3]{2}[/latex]
69. [latex]\frac{15 \sqrt[4]{125}}{\sqrt[4]{5}}[/latex]

For the following exercises, simplify each expression.

70. [latex]\sqrt{400x^4}[/latex]
71. [latex]\sqrt{49p}[/latex]
72. [latex]m^{\frac{5}{2}}\sqrt{289}[/latex]
73. [latex]3\sqrt{ab^2} - b \sqrt{a}[/latex]
74. [latex]\sqrt{\frac{225x^3}{49x}}[/latex]
75. [latex]\sqrt{50y^8}[/latex]
76. [latex]\sqrt{\frac{32}{14d}}[/latex]
77. [latex]\frac{\sqrt{8}}{1-\sqrt{3}}[/latex]
78. [latex]w^{3/2}\sqrt{32}-w^{3/2}\sqrt{50}[/latex]
79. [latex]\frac{\sqrt{12x}}{2+2\sqrt{3}}[/latex]
80. [latex]\sqrt{125n^{10}}[/latex]
81. [latex]\sqrt{\frac{81m}{361m^2}}[/latex]
82. [latex]\sqrt{\frac{144}{324d^2}}[/latex]
83. [latex]\sqrt[4]{\frac{162x^6}{16x^4}}[/latex]
84. [latex]\sqrt[3]{128z^3} - \sqrt[3]{-16z^3}[/latex]