{"id":2013,"date":"2020-08-21T19:42:02","date_gmt":"2020-08-21T23:42:02","guid":{"rendered":"https:\/\/opentextbc.ca\/introalgebra\/chapter\/chapter-review\/"},"modified":"2025-09-15T18:26:34","modified_gmt":"2025-09-15T22:26:34","slug":"chapter-review","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/introalgebra\/chapter\/chapter-review\/","title":{"raw":"7.6 Chapter Review","rendered":"7.6 Chapter Review"},"content":{"raw":"[latexpage]\n

Review Exercises<\/h1>\n

Simplify Expressions with Exponents<\/h2>\n

In the following exercises, simplify.<\/p>\n\n\n\n\n\n\n
1. \\({17}^{1}\\)<\/td>\n2. \\({10}^{4}\\)<\/td>\n<\/tr>\n
3. \\({\\left(0.5\\right)}^{3}\\)<\/td>\n4. \\({\\left(\\frac{2}{9}\\right)}^{2}\\)<\/td>\n<\/tr>\n
5. \\(-{2}^{6}\\)<\/td>\n6. \\({\\left(-2\\right)}^{6}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Simplify Expressions Using the Product Property for Exponents<\/h2>\nIn the following exercises, simplify each expression.\n<\/strong>\n\n\n\n\n\n
7. \\({p}^{15}\\cdot{p}^{16}\\)<\/td>\n8. \\({x}^{4}\\cdot{x}^{3}\\)<\/td>\n<\/tr>\n
9. \\(8\\cdot{8}^{5}\\)<\/td>\n10. \\({4}^{10}\\cdot{4}^{6}\\)<\/td>\n<\/tr>\n
11. \\({y}^{c}\\cdot{y}^{3}\\)<\/td>\n12. \\(n\\cdot{n}^{2}\\cdot{n}^{4}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Simplify Expressions Using the Power Property for Exponents<\/h2>\n

In the following exercises, simplify each expression.<\/p>\n\n\n\n\n\n
13. \\({\\left({5}^{3}\\right)}^{2}\\)<\/td>\n14. \\({\\left({m}^{3}\\right)}^{5}\\)<\/td>\n<\/tr>\n
15. \\({\\left({3}^{r}\\right)}^{s}\\)<\/td>\n16. \\({\\left({y}^{4}\\right)}^{x}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Simplify Expressions Using the Product to a Power Property<\/h2>\n

In the following exercises, simplify each expression.<\/p>\n\n\n\n\n\n
17. \\({\\left(-5y\\right)}^{3}\\)<\/td>\n18 \\({\\left(4a\\right)}^{2}\\)<\/td>\n<\/tr>\n
19. \\({\\left(10xyz\\right)}^{3}\\)<\/td>\n20. \\({\\left(2mn\\right)}^{5}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Simplify Expressions by Applying Several Properties<\/h2>\n

In the following exercises, simplify each expression.<\/p>\n\n\n\n\n\n\n
21. \\({\\left(4{a}^{3}{b}^{2}\\right)}^{3}\\)<\/td>\n22. \\({\\left({p}^{2}\\right)}^{5}\\cdot{\\left({p}^{3}\\right)}^{6}\\)<\/td>\n<\/tr>\n
23. \\({\\left(2{q}^{3}\\right)}^{4}{\\left(3q\\right)}^{2}\\)<\/td>\n24. \\({\\left(5x\\right)}^{2}\\left(7x\\right)\\)<\/td>\n<\/tr>\n
25. \\({\\left(\\frac{2}{5}{m}^{2}n\\right)}^{3}\\)<\/td>\n26. \\({\\left(\\frac{1}{3}{x}^{2}\\right)}^{2}{\\left(\\frac{1}{2}x\\right)}^{3}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
\n

Simplify Expressions Using the Quotient Property for Exponents<\/h2>\n

In the following exercises, simplify.<\/p>\n\n\n\n\n\n\n
27. \\(\\frac{{10}^{25}}{{10}^{5}}\\)<\/td>\n28. \\(\\frac{{u}^{24}}{{u}^{6}}\\)<\/td>\n<\/tr>\n
29. \\(\\frac{{v}^{12}}{{v}^{48}}\\)<\/td>\n30. \\(\\frac{{3}^{4}}{{3}^{6}}\\)<\/td>\n<\/tr>\n
31. \\(\\frac{5}{{5}^{8}}\\)<\/td>\n32. \\(\\frac{x}{{x}^{5}}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Simplify Expressions with Zero Exponents<\/h2>\n

In the following exercises, simplify.<\/p>\n\n\n\n\n\n\n\n
33. \\({x}^{0}\\)<\/td>\n34. \\({75}^{0}\\)<\/td>\n<\/tr>\n
35. \\(\\left(-{12}^{0}\\right)\\)\\({\\left(-12\\right)}^{0}\\)<\/td>\n36. \\(-{12}^{0}\\)<\/td>\n<\/tr>\n
37. \\({\\left(25x\\right)}^{0}\\)<\/td>\n38. \\(25{x}^{0}\\)<\/td>\n<\/tr>\n
39. \\({\\left(19n\\right)}^{0}-{\\left(25m\\right)}^{0}\\)<\/td>\n40. \\(19{n}^{0}-25{m}^{0}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Simplify Expressions Using the Quotient to a Power Property<\/h2>\n

In the following exercises, simplify.<\/p>\n\n\n\n\n\n
41. \\({\\left(\\frac{m}{3}\\right)}^{4}\\)<\/td>\n42. \\({\\left(\\frac{2}{5}\\right)}^{3}\\)<\/td>\n<\/tr>\n
43. \\({\\left(\\frac{x}{2y}\\right)}^{6}\\)<\/td>\n44. \\({\\left(\\frac{r}{s}\\right)}^{8}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Simplify Expressions by Applying Several Properties<\/h2>\n

In the following exercises, simplify.<\/p>\n\n\n\n\n\n\n\n
45. \\(\\frac{{n}^{10}}{{\\left({n}^{5}\\right)}^{2}}\\)<\/td>\n46. \\(\\frac{{\\left({x}^{3}\\right)}^{5}}{{x}^{9}}\\)<\/td>\n<\/tr>\n
47. \\({\\left(\\frac{{r}^{8}}{{r}^{3}}\\right)}^{4}\\)<\/td>\n48. \\({\\left(\\frac{{q}^{6}}{{q}^{8}}\\right)}^{3}\\)<\/td>\n<\/tr>\n
49. \\({\\left(\\frac{3{x}^{4}}{2{y}^{2}}\\right)}^{5}\\)<\/td>\n50. \\({\\left(\\frac{{c}^{2}}{{d}^{5}}\\right)}^{9}\\)<\/td>\n<\/tr>\n
51. \\(\\frac{{\\left(3{n}^{2}\\right)}^{4}{\\left(-5{n}^{4}\\right)}^{3}}{{\\left(-2{n}^{5}\\right)}^{2}}\\)<\/td>\n52. \\({\\left(\\frac{{v}^{3}{v}^{9}}{{v}^{6}}\\right)}^{4}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Divide Monomials<\/h2>\n

In the following exercises, divide the monomials.<\/p>\n\n\n\n\n\n
53. \\(\\frac{64{a}^{5}{b}^{9}}{-16{a}^{10}{b}^{3}}\\)<\/td>\n54. \\(-65{y}^{14}\\)\u00f7 \\(5{y}^{2}\\)<\/td>\n<\/tr>\n
55. \\(\\frac{\\left(8{p}^{6}{q}^{2}\\right)\\left(9{p}^{3}{q}^{5}\\right)}{16{p}^{8}{q}^{7}}\\)<\/td>\n56. \\(\\frac{144{x}^{15}{y}^{8}{z}^{3}}{18{x}^{10}{y}^{2}{z}^{12}}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
\n

Use the Definition of a Negative Exponent<\/h2>\n

In the following exercises, simplify.<\/p>\n\n\n\n\n\n\n
57. \\({\\left(-5\\right)}^{-3}\\)<\/td>\n58. \\({9}^{-2}\\)<\/td>\n<\/tr>\n
59. \\({\\left(6u\\right)}^{-3}\\)<\/td>\n60. \\(3\\cdot{4}^{-3}\\)<\/td>\n<\/tr>\n
61. \\({\\left(\\frac{3}{4}\\right)}^{-2}\\)<\/td>\n62. \\({\\left(\\frac{2}{5}\\right)}^{-1}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Simplify Expressions with Integer Exponents<\/h2>\n

In the following exercises, simplify.<\/p>\n\n\n\n\n\n\n\n
63. \\({q}^{-6}\\cdot{q}^{-5}\\)<\/td>\n64. \\({p}^{-2}\\cdot{p}^{8}\\)<\/td>\n<\/tr>\n
65. \\({\\left({y}^{8}\\right)}^{-1}\\)<\/td>\n66. \\(\\left({c}^{-2}d\\right)\\left({c}^{-3}{d}^{-2}\\right)\\)<\/td>\n<\/tr>\n
67. \\(\\frac{{a}^{8}}{{a}^{12}}\\)<\/td>\n68. \\({\\left({q}^{-4}\\right)}^{-3}\\)<\/td>\n<\/tr>\n
69. \\(\\frac{{r}^{-2}}{{r}^{-3}}\\)<\/td>\n70. \\(\\frac{{n}^{5}}{{n}^{-4}}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Convert from Decimal Notation to Scientific Notation<\/h2>\n

In the following exercises, write each number in scientific notation.<\/p>\n\n\n\n\n\n
71. 0.00429<\/td>\n72. 8,500,000<\/td>\n<\/tr>\n
73. In 2015, the population of the world was about 7,200,000,000 people.<\/td>\n74. The thickness of a dime is about 0.053 inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Convert Scientific Notation to Decimal Form<\/h2>\n

In the following exercises, convert each number to decimal form.<\/p>\n\n\n\n\n\n
75. 1.5 \u00d7\\({10}^{10}\\)<\/td>\n76. 3.8 \u00d7\\({10}^{5}\\)<\/td>\n<\/tr>\n
77.\u00a0 5.5 \u00d7\\({10}^{-1}\\)<\/td>\n78. 9.1 \u00d7\\({10}^{-7}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Multiply and Divide Using Scientific Notation<\/h2>\n

In the following exercises, multiply and write your answer in decimal form.<\/p>\n\n\n\n\n
79. 3.5 \u00d7\\({10}^{-2})\\)(6.2 \u00d7 \\({10}^{-1})\\)<\/td>\n80.\u00a0 2 \u00d7\\({10}^{5})\\) (4 \u00d7\\({10}^{-3})\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nIn the following exercises, divide and write your answer in decimal form.\n\n<\/div>\n
\n\n\n\n
81. \\(\\frac{9\\phantom{\\rule{0.2em}{0ex}}\\times \\phantom{\\rule{0.2em}{0ex}}{10}^{-5}}{3\\phantom{\\rule{0.2em}{0ex}}\\times \\phantom{\\rule{0.2em}{0ex}}{10}^{2}}\\)<\/td>\n82. \\(\\frac{8\\phantom{\\rule{0.2em}{0ex}}\\times \\phantom{\\rule{0.2em}{0ex}}{10}^{5}}{4\\phantom{\\rule{0.2em}{0ex}}\\times \\phantom{\\rule{0.2em}{0ex}}{10}^{-1}}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
\n

Simplify Expressions with Square Roots<\/h2>\n

In the following exercises, simplify.<\/p>\n\n\n\n\n
83.\\(\\sqrt{144}\\)\n\n85. \\(-\\sqrt{81}\\)\n\n87. \\(\\sqrt{-36}\\)\n\n89. \\(\\sqrt{64+225}\\)<\/td>\n84. \\(\\sqrt{64}\\)\n\n86. \\(-\\sqrt{25}\\)\n\n88. \\(\\sqrt{-9}\\)\n\n90. \\(\\sqrt{64}+\\sqrt{225}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
\n

Estimate Square Roots<\/h2>\n<\/div>\n<\/div>\n

In the following exercises, estimate each square root between two consecutive whole numbers.<\/p>\n\n

\n
\n\n\n\n
91.\\(\\sqrt{155}\\)<\/td>\n92. \\(\\sqrt{28}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

Approximate Square Roots<\/h2>\n

In the following exercises, approximate each square root and round to two decimal places.<\/p>\n\n

\n
\n\n\n\n
93. \\(\\sqrt{57}\\)<\/td>\n94. \\(\\sqrt{15}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

Simplify Variable Expressions with Square Roots<\/h2>\n

In the following exercises, simplify. (Assume all variables are greater than or equal to zero.)<\/p>\n\n

\n
\n\n\n\n
95.\\(\\sqrt{64{b}^{2}}\\)\n\n97. \\(\\sqrt{225{m}^{2}{n}^{2}}\\)\n\n99. \\(\\sqrt{49{y}^{2}}\\)\n\n101. \\(\\sqrt{121{c}^{2}{d}^{2}}\\)<\/td>\n96.\\(\\sqrt{{q}^{2}}\\)\n\n98. \\(-\\sqrt{121{a}^{2}}\\)\n\n100. \\(-\\sqrt{100{q}^{2}}\\)\n\n102. \\(\\sqrt{4{a}^{2}{b}^{2}}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

Use Square Roots in Applications<\/h2>\n

In the following exercises, solve. Round to one decimal place.<\/p>\n\n

\n
\n\n\n\n
103.Landscaping<\/strong> Janet wants to plant a square flower garden in her yard. She has enough topsoil to cover an area of \\(30\\) square feet. How long can a side of the flower garden be?\n\n105. Accident investigation<\/strong> The skid marks of a car involved in an accident were \\(216\\) feet. How fast had the car been going before applying the brakes?<\/td>\n104. Art<\/strong> Diego has \\(225\\) square inch tiles. He wants to use them to make a square mosaic. How long can each side of the mosaic be?\n\n106. Gravity<\/strong> A hiker dropped a granola bar from a lookout spot \\(576\\) feet above a valley. How long did it take the granola bar to reach the valley floor?<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Review Exercise Answers<\/h1>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
1. 17<\/td>\n3. 0.125<\/td>\n<\/tr>\n
5. \\(-64\\)<\/td>\n7. \\({p}^{31}\\)<\/td>\n<\/tr>\n
9. \\({8}^{6}\\)<\/td>\n11. \\({y}^{c+3}\\)<\/td>\n<\/tr>\n
13. \\({5}^{6}\\)<\/td>\n15. \\({3}^{rs}\\)<\/td>\n<\/tr>\n
17. \\(-125{y}^{3}\\)<\/td>\n19. \\(1000{x}^{3}{y}^{3}{z}^{3}\\)<\/td>\n<\/tr>\n
21. \\(64{a}^{9}{b}^{6}\\)<\/td>\n23. \\(48{q}^{14}\\)<\/td>\n<\/tr>\n
25. \\(\\frac{8}{125}{m}^{6}{n}^{3}\\)<\/td>\n27. \\({10}^{20}\\)<\/td>\n<\/tr>\n
29. \\(\\frac{1}{{v}^{36}}\\)<\/td>\n31. \\(\\frac{1}{{5}^{7}}\\)<\/td>\n<\/tr>\n
33. 1<\/td>\n35. 1<\/td>\n<\/tr>\n
37. 1<\/td>\n39. 0<\/td>\n<\/tr>\n
41. \\(\\frac{{m}^{4}}{81}\\)<\/td>\n43. \\(\\frac{{x}^{6}}{64{y}^{6}}\\)<\/td>\n<\/tr>\n
45. 1<\/td>\n47. \\({r}^{20}\\)<\/td>\n<\/tr>\n
49. \\(\\frac{343{x}^{20}}{32{y}^{10}}\\)<\/td>\n51. \\(-\\frac{10,125{n}^{10}}{4}\\)<\/td>\n<\/tr>\n
53. \\(-\\frac{4{b}^{6}}{{a}^{5}}\\)<\/td>\n55. \\(\\frac{9p}{2}\\)<\/td>\n<\/tr>\n
57. \\(-\\frac{1}{125}\\)<\/td>\n59.\\(\\frac{1}{216{u}^{3}}\\)<\/td>\n<\/tr>\n
61. \\(\\frac{16}{9}\\)<\/td>\n63. \\(\\frac{1}{{q}^{11}}\\)<\/td>\n<\/tr>\n
65. \\(\\frac{1}{{y}^{8}}\\)<\/td>\n67. \\(\\frac{1}{{a}^{4}}\\)<\/td>\n<\/tr>\n
69. \\(r\\)<\/td>\n71. \\(4.29\\phantom{\\rule{0.2em}{0ex}}\\times \\phantom{\\rule{0.2em}{0ex}}{10}^{-3}\\)<\/td>\n<\/tr>\n
73. \\(7.2\\phantom{\\rule{0.2em}{0ex}}\\times \\phantom{\\rule{0.2em}{0ex}}{10}^{9}\\)<\/td>\n75. \\(15,000,000,000\\)<\/td>\n<\/tr>\n
77. \\(0.55\\)<\/td>\n79. \\(0.0217\\)<\/td>\n<\/tr>\n
81. \\(0.0000003\\)<\/td>\n83. 12<\/td>\n<\/tr>\n
85. \u22129<\/td>\n87. not a real number<\/td>\n<\/tr>\n
89. 17<\/td>\n91. \\(12<\\sqrt{155}<13\\)<\/td>\n<\/tr>\n
93. 7.55<\/td>\n95. 8b<\/td>\n<\/tr>\n
97. 15mn<\/td>\n99. 7y<\/td>\n<\/tr>\n
101. 11cd<\/td>\n103. 5.5 feet<\/td>\n<\/tr>\n
105. 72 mph<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Practice Test<\/h1>\nIn the following exercises, simplify each expression.\n\n\n\n\n\n\n\n\n\n\n\n\n\n
1. \\({\\left(-\\frac{2}{5}\\right)}^{3}\\)<\/td>\n2.\u00a0 \u00a0 \\(u\\cdot{u}^{4}\\)<\/td>\n<\/tr>\n
3.\u00a0 \\({\\left(4{a}^{3}{b}^{5}\\right)}^{2}\\)<\/td>\n4. \\(\\frac{{n}^{-2}}{{n}^{-10}}\\)<\/td>\n<\/tr>\n
5. \\(\\frac{{3}^{8}}{{3}^{10}}\\)<\/td>\n6. \\({\\left(\\frac{{v}^{2}{v}^{6}}{{v}^{4}}\\right)}^{2}\\)<\/td>\n<\/tr>\n
7. \\({\\left(87{x}^{15}{y}^{3}{z}^{22}\\right)}^{0}\\)<\/td>\n8. \\({\\left(\\frac{{m}^{4}\\cdot m}{{m}^{3}}\\right)}^{6}\\)<\/td>\n<\/tr>\n
9. \\(\\frac{80{c}^{8}{d}^{2}}{16c{d}^{10}}\\)<\/td>\n10. \\({5}^{-2}\\)<\/td>\n<\/tr>\n
11. \\({q}^{-4}\\cdot{q}^{-5}\\)<\/td>\n12. \\({\\left(4m\\right)}^{-3}\\)<\/td>\n<\/tr>\n
13. \\(\\frac{8.4\\phantom{\\rule{0.2em}{0ex}}\\times \\phantom{\\rule{0.2em}{0ex}}{10}^{-3}}{4\\phantom{\\rule{0.2em}{0ex}}\\times \\phantom{\\rule{0.2em}{0ex}}{10}^{3}}\\)<\/td>\n14. \\(\\left(3.4\\phantom{\\rule{0.2em}{0ex}}\\times \\phantom{\\rule{0.2em}{0ex}}{10}^{9}\\right)\\left(2.2\\phantom{\\rule{0.2em}{0ex}}\\times \\phantom{\\rule{0.2em}{0ex}}{10}^{-5}\\right)\\)<\/td>\n<\/tr>\n
15. \\(\\sqrt{81}\\)<\/td>\n16. \\(-\\sqrt{49}\\)<\/td>\n<\/tr>\n
17. \\(\\sqrt{-16}\\)<\/td>\n18. \\(\\sqrt{{b}^{2}}\\)<\/td>\n<\/tr>\n
19. \\(-\\sqrt{64{a}^{2}}\\)<\/td>\n20. \\(-\\sqrt{144{q}^{2}}\\)<\/td>\n<\/tr>\n
21. Convert 83,000,000 to scientific notation.<\/td>\n22. Convert \\(6.91\\phantom{\\rule{0.2em}{0ex}}\\times \\phantom{\\rule{0.2em}{0ex}}{10}^{-5}\\) to decimal form.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
<\/div>\n<\/div>\n

Practice Test Answers<\/h1>\n\n\n\n\n\n\n\n\n\n\n\n\n\n
1. \\(-\\frac{8}{125}\\)<\/td>\n2. \\({u}^{5}\\)<\/td>\n<\/tr>\n
3. \\(16{a}^{6}{b}^{10}\\)<\/td>\n4. \\({n}^{8}\\)<\/td>\n<\/tr>\n
\u00a05. \\(\\frac{1}{9}\\)<\/td>\n6. \\({v}^{8}\\)<\/td>\n<\/tr>\n
7. \\(1\\)<\/td>\n8. \\({m}^{12}\\)<\/td>\n<\/tr>\n
\u00a09. \\(\\frac{5{c}^7}{{d}^{8}}\\)<\/td>\n\u00a010. \\(\\frac{1}{25}\\)<\/td>\n<\/tr>\n
\u00a011. \\(\\frac{1}{q}\\)<\/td>\n\u00a012. \\(\\frac{1}{64 {m}^{3}}\\)<\/td>\n<\/tr>\n
13. \\(2.1\\times {10}^{-6}\\)<\/td>\n14. \\(7.48\\times {10}^{4}\\)<\/td>\n<\/tr>\n
15. 9<\/td>\n16. -7<\/td>\n<\/tr>\n
17. not a real number<\/td>\n18. \\(b\\)<\/td>\n<\/tr>\n
19. \\(-8a\\)<\/td>\n20. \\(-12q\\)<\/td>\n<\/tr>\n
21. \\(8.3\\phantom{\\rule{0.2em}{0ex}}\\times \\phantom{\\rule{0.2em}{0ex}}{10}^{7}\\)<\/td>\n22. 0.0000691<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","rendered":"

Review Exercises<\/h1>\n

Simplify Expressions with Exponents<\/h2>\n

In the following exercises, simplify.<\/p>\n\n\n\n\n\n
1. \"{17}^{1}\"<\/td>\n2. \"{10}^{4}\"<\/td>\n<\/tr>\n
3. \"{\left(0.5\right)}^{3}\"<\/td>\n4. \"{\left(\frac{2}{9}\right)}^{2}\"<\/td>\n<\/tr>\n
5. \"-{2}^{6}\"<\/td>\n6. \"{\left(-2\right)}^{6}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Simplify Expressions Using the Product Property for Exponents<\/h2>\n

In the following exercises, simplify each expression.
\n<\/strong><\/p>\n\n\n\n\n\n
7. \"{p}^{15}\cdot{p}^{16}\"<\/td>\n8. \"{x}^{4}\cdot{x}^{3}\"<\/td>\n<\/tr>\n
9. \"8\cdot{8}^{5}\"<\/td>\n10. \"{4}^{10}\cdot{4}^{6}\"<\/td>\n<\/tr>\n
11. \"{y}^{c}\cdot{y}^{3}\"<\/td>\n12. \"n\cdot{n}^{2}\cdot{n}^{4}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Simplify Expressions Using the Power Property for Exponents<\/h2>\n

In the following exercises, simplify each expression.<\/p>\n\n\n\n\n
13. \"{\left({5}^{3}\right)}^{2}\"<\/td>\n14. \"{\left({m}^{3}\right)}^{5}\"<\/td>\n<\/tr>\n
15. \"{\left({3}^{r}\right)}^{s}\"<\/td>\n16. \"{\left({y}^{4}\right)}^{x}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Simplify Expressions Using the Product to a Power Property<\/h2>\n

In the following exercises, simplify each expression.<\/p>\n\n\n\n\n
17. \"{\left(-5y\right)}^{3}\"<\/td>\n18 \"{\left(4a\right)}^{2}\"<\/td>\n<\/tr>\n
19. \"{\left(10xyz\right)}^{3}\"<\/td>\n20. \"{\left(2mn\right)}^{5}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Simplify Expressions by Applying Several Properties<\/h2>\n

In the following exercises, simplify each expression.<\/p>\n\n\n\n\n\n
21. \"{\left(4{a}^{3}{b}^{2}\right)}^{3}\"<\/td>\n22. \"{\left({p}^{2}\right)}^{5}\cdot{\left({p}^{3}\right)}^{6}\"<\/td>\n<\/tr>\n
23. \"{\left(2{q}^{3}\right)}^{4}{\left(3q\right)}^{2}\"<\/td>\n24. \"{\left(5x\right)}^{2}\left(7x\right)\"<\/td>\n<\/tr>\n
25. \"{\left(\frac{2}{5}{m}^{2}n\right)}^{3}\"<\/td>\n26. \"{\left(\frac{1}{3}{x}^{2}\right)}^{2}{\left(\frac{1}{2}x\right)}^{3}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
\n

Simplify Expressions Using the Quotient Property for Exponents<\/h2>\n

In the following exercises, simplify.<\/p>\n\n\n\n\n\n
27. \"\frac{{10}^{25}}{{10}^{5}}\"<\/td>\n28. \"\frac{{u}^{24}}{{u}^{6}}\"<\/td>\n<\/tr>\n
29. \"\frac{{v}^{12}}{{v}^{48}}\"<\/td>\n30. \"\frac{{3}^{4}}{{3}^{6}}\"<\/td>\n<\/tr>\n
31. \"\frac{5}{{5}^{8}}\"<\/td>\n32. \"\frac{x}{{x}^{5}}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Simplify Expressions with Zero Exponents<\/h2>\n

In the following exercises, simplify.<\/p>\n\n\n\n\n\n\n
33. \"{x}^{0}\"<\/td>\n34. \"{75}^{0}\"<\/td>\n<\/tr>\n
35. \"\left(-{12}^{0}\right)\"\"{\left(-12\right)}^{0}\"<\/td>\n36. \"-{12}^{0}\"<\/td>\n<\/tr>\n
37. \"{\left(25x\right)}^{0}\"<\/td>\n38. \"25{x}^{0}\"<\/td>\n<\/tr>\n
39. \"{\left(19n\right)}^{0}-{\left(25m\right)}^{0}\"<\/td>\n40. \"19{n}^{0}-25{m}^{0}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Simplify Expressions Using the Quotient to a Power Property<\/h2>\n

In the following exercises, simplify.<\/p>\n\n\n\n\n
41. \"{\left(\frac{m}{3}\right)}^{4}\"<\/td>\n42. \"{\left(\frac{2}{5}\right)}^{3}\"<\/td>\n<\/tr>\n
43. \"{\left(\frac{x}{2y}\right)}^{6}\"<\/td>\n44. \"{\left(\frac{r}{s}\right)}^{8}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Simplify Expressions by Applying Several Properties<\/h2>\n

In the following exercises, simplify.<\/p>\n\n\n\n\n\n\n
45. \"\frac{{n}^{10}}{{\left({n}^{5}\right)}^{2}}\"<\/td>\n46. \"\frac{{\left({x}^{3}\right)}^{5}}{{x}^{9}}\"<\/td>\n<\/tr>\n
47. \"{\left(\frac{{r}^{8}}{{r}^{3}}\right)}^{4}\"<\/td>\n48. \"{\left(\frac{{q}^{6}}{{q}^{8}}\right)}^{3}\"<\/td>\n<\/tr>\n
49. \"{\left(\frac{3{x}^{4}}{2{y}^{2}}\right)}^{5}\"<\/td>\n50. \"{\left(\frac{{c}^{2}}{{d}^{5}}\right)}^{9}\"<\/td>\n<\/tr>\n
51. \"\frac{{\left(3{n}^{2}\right)}^{4}{\left(-5{n}^{4}\right)}^{3}}{{\left(-2{n}^{5}\right)}^{2}}\"<\/td>\n52. \"{\left(\frac{{v}^{3}{v}^{9}}{{v}^{6}}\right)}^{4}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Divide Monomials<\/h2>\n

In the following exercises, divide the monomials.<\/p>\n\n\n\n\n
53. \"\frac{64{a}^{5}{b}^{9}}{-16{a}^{10}{b}^{3}}\"<\/td>\n54. \"-65{y}^{14}\"\u00f7 \"5{y}^{2}\"<\/td>\n<\/tr>\n
55. \"\frac{\left(8{p}^{6}{q}^{2}\right)\left(9{p}^{3}{q}^{5}\right)}{16{p}^{8}{q}^{7}}\"<\/td>\n56. \"\frac{144{x}^{15}{y}^{8}{z}^{3}}{18{x}^{10}{y}^{2}{z}^{12}}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
\n

Use the Definition of a Negative Exponent<\/h2>\n

In the following exercises, simplify.<\/p>\n\n\n\n\n\n
57. \"{\left(-5\right)}^{-3}\"<\/td>\n58. \"{9}^{-2}\"<\/td>\n<\/tr>\n
59. \"{\left(6u\right)}^{-3}\"<\/td>\n60. \"3\cdot{4}^{-3}\"<\/td>\n<\/tr>\n
61. \"{\left(\frac{3}{4}\right)}^{-2}\"<\/td>\n62. \"{\left(\frac{2}{5}\right)}^{-1}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Simplify Expressions with Integer Exponents<\/h2>\n

In the following exercises, simplify.<\/p>\n\n\n\n\n\n\n
63. \"{q}^{-6}\cdot{q}^{-5}\"<\/td>\n64. \"{p}^{-2}\cdot{p}^{8}\"<\/td>\n<\/tr>\n
65. \"{\left({y}^{8}\right)}^{-1}\"<\/td>\n66. \"\left({c}^{-2}d\right)\left({c}^{-3}{d}^{-2}\right)\"<\/td>\n<\/tr>\n
67. \"\frac{{a}^{8}}{{a}^{12}}\"<\/td>\n68. \"{\left({q}^{-4}\right)}^{-3}\"<\/td>\n<\/tr>\n
69. \"\frac{{r}^{-2}}{{r}^{-3}}\"<\/td>\n70. \"\frac{{n}^{5}}{{n}^{-4}}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Convert from Decimal Notation to Scientific Notation<\/h2>\n

In the following exercises, write each number in scientific notation.<\/p>\n\n\n\n\n
71. 0.00429<\/td>\n72. 8,500,000<\/td>\n<\/tr>\n
73. In 2015, the population of the world was about 7,200,000,000 people.<\/td>\n74. The thickness of a dime is about 0.053 inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Convert Scientific Notation to Decimal Form<\/h2>\n

In the following exercises, convert each number to decimal form.<\/p>\n\n\n\n\n
75. 1.5 \u00d7\"{10}^{10}\"<\/td>\n76. 3.8 \u00d7\"{10}^{5}\"<\/td>\n<\/tr>\n
77.\u00a0 5.5 \u00d7\"{10}^{-1}\"<\/td>\n78. 9.1 \u00d7\"{10}^{-7}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Multiply and Divide Using Scientific Notation<\/h2>\n

In the following exercises, multiply and write your answer in decimal form.<\/p>\n\n\n\n
79. 3.5 \u00d7\"{10}^{-2})\"(6.2 \u00d7 \"{10}^{-1})\"<\/td>\n80.\u00a0 2 \u00d7\"{10}^{5})\" (4 \u00d7\"{10}^{-3})\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

In the following exercises, divide and write your answer in decimal form.<\/p>\n<\/div>\n

\n\n\n\n
81. \"\frac{9\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-5}}{3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{2}}\"<\/td>\n82. \"\frac{8\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}}{4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-1}}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
\n

Simplify Expressions with Square Roots<\/h2>\n

In the following exercises, simplify.<\/p>\n\n\n\n
83.\"\sqrt{144}\"<\/p>\n

85. \"-\sqrt{81}\"<\/p>\n

87. \"\sqrt{-36}\"<\/p>\n

89. \"\sqrt{64+225}\"<\/td>\n

84. \"\sqrt{64}\"<\/p>\n

86. \"-\sqrt{25}\"<\/p>\n

88. \"\sqrt{-9}\"<\/p>\n

90. \"\sqrt{64}+\sqrt{225}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\n
\n

Estimate Square Roots<\/h2>\n<\/div>\n<\/div>\n

In the following exercises, estimate each square root between two consecutive whole numbers.<\/p>\n

\n
\n\n\n\n
91.\"\sqrt{155}\"<\/td>\n92. \"\sqrt{28}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

Approximate Square Roots<\/h2>\n

In the following exercises, approximate each square root and round to two decimal places.<\/p>\n

\n
\n\n\n\n
93. \"\sqrt{57}\"<\/td>\n94. \"\sqrt{15}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

Simplify Variable Expressions with Square Roots<\/h2>\n

In the following exercises, simplify. (Assume all variables are greater than or equal to zero.)<\/p>\n

\n
\n\n\n\n
95.\"\sqrt{64{b}^{2}}\"<\/p>\n

97. \"\sqrt{225{m}^{2}{n}^{2}}\"<\/p>\n

99. \"\sqrt{49{y}^{2}}\"<\/p>\n

101. \"\sqrt{121{c}^{2}{d}^{2}}\"<\/td>\n

96.\"\sqrt{{q}^{2}}\"<\/p>\n

98. \"-\sqrt{121{a}^{2}}\"<\/p>\n

100. \"-\sqrt{100{q}^{2}}\"<\/p>\n

102. \"\sqrt{4{a}^{2}{b}^{2}}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

Use Square Roots in Applications<\/h2>\n

In the following exercises, solve. Round to one decimal place.<\/p>\n

\n
\n\n\n\n
103.Landscaping<\/strong> Janet wants to plant a square flower garden in her yard. She has enough topsoil to cover an area of \"30\" square feet. How long can a side of the flower garden be?<\/p>\n

105. Accident investigation<\/strong> The skid marks of a car involved in an accident were \"216\" feet. How fast had the car been going before applying the brakes?<\/td>\n

104. Art<\/strong> Diego has \"225\" square inch tiles. He wants to use them to make a square mosaic. How long can each side of the mosaic be?<\/p>\n

106. Gravity<\/strong> A hiker dropped a granola bar from a lookout spot \"576\" feet above a valley. How long did it take the granola bar to reach the valley floor?<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Review Exercise Answers<\/h1>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
1. 17<\/td>\n3. 0.125<\/td>\n<\/tr>\n
5. \"-64\"<\/td>\n7. \"{p}^{31}\"<\/td>\n<\/tr>\n
9. \"{8}^{6}\"<\/td>\n11. \"{y}^{c+3}\"<\/td>\n<\/tr>\n
13. \"{5}^{6}\"<\/td>\n15. \"{3}^{rs}\"<\/td>\n<\/tr>\n
17. \"-125{y}^{3}\"<\/td>\n19. \"1000{x}^{3}{y}^{3}{z}^{3}\"<\/td>\n<\/tr>\n
21. \"64{a}^{9}{b}^{6}\"<\/td>\n23. \"48{q}^{14}\"<\/td>\n<\/tr>\n
25. \"\frac{8}{125}{m}^{6}{n}^{3}\"<\/td>\n27. \"{10}^{20}\"<\/td>\n<\/tr>\n
29. \"\frac{1}{{v}^{36}}\"<\/td>\n31. \"\frac{1}{{5}^{7}}\"<\/td>\n<\/tr>\n
33. 1<\/td>\n35. 1<\/td>\n<\/tr>\n
37. 1<\/td>\n39. 0<\/td>\n<\/tr>\n
41. \"\frac{{m}^{4}}{81}\"<\/td>\n43. \"\frac{{x}^{6}}{64{y}^{6}}\"<\/td>\n<\/tr>\n
45. 1<\/td>\n47. \"{r}^{20}\"<\/td>\n<\/tr>\n
49. \"\frac{343{x}^{20}}{32{y}^{10}}\"<\/td>\n51. \"-\frac{10,125{n}^{10}}{4}\"<\/td>\n<\/tr>\n
53. \"-\frac{4{b}^{6}}{{a}^{5}}\"<\/td>\n55. \"\frac{9p}{2}\"<\/td>\n<\/tr>\n
57. \"-\frac{1}{125}\"<\/td>\n59.\"\frac{1}{216{u}^{3}}\"<\/td>\n<\/tr>\n
61. \"\frac{16}{9}\"<\/td>\n63. \"\frac{1}{{q}^{11}}\"<\/td>\n<\/tr>\n
65. \"\frac{1}{{y}^{8}}\"<\/td>\n67. \"\frac{1}{{a}^{4}}\"<\/td>\n<\/tr>\n
69. \"r\"<\/td>\n71. \"4.29\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-3}\"<\/td>\n<\/tr>\n
73. \"7.2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{9}\"<\/td>\n75. \"15,000,000,000\"<\/td>\n<\/tr>\n
77. \"0.55\"<\/td>\n79. \"0.0217\"<\/td>\n<\/tr>\n
81. \"0.0000003\"<\/td>\n83. 12<\/td>\n<\/tr>\n
85. \u22129<\/td>\n87. not a real number<\/td>\n<\/tr>\n
89. 17<\/td>\n91. \"12<\sqrt{155}<13\"<\/td>\n<\/tr>\n
93. 7.55<\/td>\n95. 8b<\/td>\n<\/tr>\n
97. 15mn<\/td>\n99. 7y<\/td>\n<\/tr>\n
101. 11cd<\/td>\n103. 5.5 feet<\/td>\n<\/tr>\n
105. 72 mph<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Practice Test<\/h1>\n

In the following exercises, simplify each expression.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n
1. \"{\left(-\frac{2}{5}\right)}^{3}\"<\/td>\n2.\u00a0 \u00a0 \"u\cdot{u}^{4}\"<\/td>\n<\/tr>\n
3.\u00a0 \"{\left(4{a}^{3}{b}^{5}\right)}^{2}\"<\/td>\n4. \"\frac{{n}^{-2}}{{n}^{-10}}\"<\/td>\n<\/tr>\n
5. \"\frac{{3}^{8}}{{3}^{10}}\"<\/td>\n6. \"{\left(\frac{{v}^{2}{v}^{6}}{{v}^{4}}\right)}^{2}\"<\/td>\n<\/tr>\n
7. \"{\left(87{x}^{15}{y}^{3}{z}^{22}\right)}^{0}\"<\/td>\n8. \"{\left(\frac{{m}^{4}\cdot m}{{m}^{3}}\right)}^{6}\"<\/td>\n<\/tr>\n
9. \"\frac{80{c}^{8}{d}^{2}}{16c{d}^{10}}\"<\/td>\n10. \"{5}^{-2}\"<\/td>\n<\/tr>\n
11. \"{q}^{-4}\cdot{q}^{-5}\"<\/td>\n12. \"{\left(4m\right)}^{-3}\"<\/td>\n<\/tr>\n
13. \"\frac{8.4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-3}}{4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}}\"<\/td>\n14. \"\left(3.4\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{9}\right)\left(2.2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-5}\right)\"<\/td>\n<\/tr>\n
15. \"\sqrt{81}\"<\/td>\n16. \"-\sqrt{49}\"<\/td>\n<\/tr>\n
17. \"\sqrt{-16}\"<\/td>\n18. \"\sqrt{{b}^{2}}\"<\/td>\n<\/tr>\n
19. \"-\sqrt{64{a}^{2}}\"<\/td>\n20. \"-\sqrt{144{q}^{2}}\"<\/td>\n<\/tr>\n
21. Convert 83,000,000 to scientific notation.<\/td>\n22. Convert \"6.91\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-5}\" to decimal form.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n
<\/div>\n<\/div>\n

Practice Test Answers<\/h1>\n\n\n\n\n\n\n\n\n\n\n\n\n\n
1. \"-\frac{8}{125}\"<\/td>\n2. \"{u}^{5}\"<\/td>\n<\/tr>\n
3. \"16{a}^{6}{b}^{10}\"<\/td>\n4. \"{n}^{8}\"<\/td>\n<\/tr>\n
\u00a05. \"\frac{1}{9}\"<\/td>\n6. \"{v}^{8}\"<\/td>\n<\/tr>\n
7. \"1\"<\/td>\n8. \"{m}^{12}\"<\/td>\n<\/tr>\n
\u00a09. \"\frac{5{c}^7}{{d}^{8}}\"<\/td>\n\u00a010. \"\frac{1}{25}\"<\/td>\n<\/tr>\n
\u00a011. \"\frac{1}{q}\"<\/td>\n\u00a012. \"\frac{1}{64 {m}^{3}}\"<\/td>\n<\/tr>\n
13. \"2.1\times {10}^{-6}\"<\/td>\n14. \"7.48\times {10}^{4}\"<\/td>\n<\/tr>\n
15. 9<\/td>\n16. -7<\/td>\n<\/tr>\n
17. not a real number<\/td>\n18. \"b\"<\/td>\n<\/tr>\n
19. \"-8a\"<\/td>\n20. \"-12q\"<\/td>\n<\/tr>\n
21. \"8.3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{7}\"<\/td>\n22. 0.0000691<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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