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Mathematics LibreTexts

5.2E: Exercises

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    30852
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    Practice Makes Perfect

    Simplify Expressions Using the Properties for Exponents

    In the following exercises, simplify each expression using the properties for exponents.

    1. ⓐ \(d^3·d^6\) ⓑ \(4^{5x}·4^{9x}\) ⓒ \(2y·4y^3\) ⓓ \(w·w^2·w^3\)

    Answer

    ⓐ \(d^9\) ⓑ \(4^{14x}\) ⓒ \(8y^4\) ⓓ \(w^6\)

    2. ⓐ \(x^4·x^2\) ⓑ \(8^{9x}·8^3\) ⓒ \(3z^{25}·5z^8\) ⓓ \(y·y^3·y^5\)

    3. ⓐ \(n^{19}·n^{12}\) ⓑ \(3^x·3^6\) ⓒ \(7w^5·8w\) ⓓ \(a^4·a^3·a^9\)

    Answer

    ⓐ \(n^{31}\) ⓑ \(3^{x+6}\) ⓒ \(56w^6\)
    ⓓ \(a^{16}\)

    4. ⓐ \(q^{27}·q^{15}\) ⓑ \(5^x·5^{4x}\) ⓒ \(9u^{41}·7u^{53}\)
    ⓓ \(c^5·c^{11}·c^2\)

    5. \(m^x·m^3\)

    Answer

    \(m^{x+3}\)

    6. \(n^y·n^2\)

    7. \(y^a·y^b\)

    Answer

    \(y^{a+b}\)

    8. \(x^p·x^q\)

    9. ⓐ \(\dfrac{x^{18}}{x^3}\) ⓑ \(\dfrac{5^{12}}{5^3}\) ⓒ \(\dfrac{q^{18}}{q^{36}}\) ⓓ \(\dfrac{10^2}{10^3}\)

    Answer

    ⓐ \(x^{15}\) ⓑ \(5^9\) ⓒ \(\dfrac{1}{q^{18}}\) ⓓ \(\dfrac{1}{10}\)

    10. ⓐ \(\dfrac{y^{20}}{y^{10}}\) ⓑ \(\dfrac{7^{16}}{7^2}\) ⓒ \(\dfrac{t^{10}}{t^{40}}\) ⓓ \(\dfrac{8^3}{8^5}\)

    11. ⓐ \(\dfrac{p^{21}}{p^7}\) ⓑ \(\dfrac{4^{16}}{4^4}\) ⓒ \(\dfrac{b}{b^9}\) ⓓ \(\dfrac{4}{4^6}\)

    Answer

    ⓐ \(p^{14}\) ⓑ \(4^{12}\) ⓒ \(\dfrac{1}{b^8}\) ⓓ \(\dfrac{1}{4^5}\)

    12. ⓐ \(\dfrac{u^{24}}{u^3}\) ⓑ \(\dfrac{9^{15}}{9^5}\) ⓒ \(\dfrac{x}{x^7}\) ⓓ \(\dfrac{10}{10^3}\)

    13. ⓐ \(20^0\) ⓑ \(b^0\)

    Answer

    ⓐ 1 ⓑ 1

    14. ⓐ \(13^0\) ⓑ \(k^0\)

    15. ⓐ \(−27^0\) ⓑ \(−(27^0)\)

    Answer

    ⓐ \(−1\) ⓑ \(−1\)

    16. ⓐ \(−15^0\) ⓑ \(−(15^0)\)

    Use the Definition of a Negative Exponent

    In the following exercises, simplify each expression.

    17. ⓐ \(a^{−2}\) ⓑ \(10^{−3}\) ⓒ \(\dfrac{1}{c^{−5}}\) ⓓ \(\dfrac{1}{3^{−2}}\)

    Answer

    ⓐ \(\dfrac{1}{a^{2}}\) ⓑ \(\dfrac{1}{1000}\) ⓒ \(c^{5}\) ⓓ \(9\)

    18. ⓐ \(b^{−4}\) ⓑ \(10^{−2}\) ⓒ \(\dfrac{1}{c^{−5}}\) ⓓ \(\dfrac{1}{5^{−2}}\)

    19. ⓐ \(r^{−3}\) ⓑ \(10^{−5}\) ⓒ \(\dfrac{1}{q^{−10}}\) ⓓ \(\dfrac{1}{10^{−3}}\)

    Answer

    ⓐ \(\dfrac{1}{r3}\) ⓑ \(\dfrac{1}{100,000}\) ⓒ \(q^{10}\) ⓓ \(1,000\)

    20. ⓐ \(s^{−8}\) ⓑ \(10^{−2}\) ⓒ \(\dfrac{1}{t^{−9}}\) ⓓ \(\dfrac{1}{10^{−4}}\)

    21. ⓐ \(\left(\dfrac{5}{8}\right)^{-2}\) ⓑ \(\left(−\dfrac{b}{a}\right)^{−2}\)

    Answer

    ⓐ \(\dfrac{64}{25}\) ⓑ \(\dfrac{a^{2}}{b^{2}}\)

    22. ⓐ \(\left(\dfrac{3}{10}\right)^{−2}\) ⓑ \(\left(−\dfrac{2}{z}\right)^{−3}\)

    23. ⓐ \(\left(\dfrac{4}{9}\right)^{−3}\) ⓑ \(\left(−\dfrac{u}{v}\right)^{−5}\)

    Answer

    ⓐ \(\dfrac{729}{64}\) ⓑ \(−\dfrac{v^{5}}{u^{5}}\)

    24. ⓐ \(\left(\dfrac{7}{2}\right)^{−3}\) ⓑ \(\left(−\dfrac{3}{x}\right)^{−3}\)

    25. ⓐ \((−5)^{−2}\) ⓑ \(−5^{−2}\) ⓒ \(\left(−\dfrac{1}{5}\right)^{−2}\) ⓓ \(−\left(\dfrac{1}{5}\right)^{−2}\)

    Answer

    ⓐ \(\dfrac{1}{25}\) ⓑ \(\dfrac{1}{25}\) ⓒ \(25\) ⓓ \(−25\)

    26. ⓐ \(−5^{−3}\) ⓑ \(\left(−\dfrac{1}{5}\right)^{−3}\) ⓒ \(−\left(\dfrac{1}{5}\right)^{−3}\) ⓓ \((−5)^{−3}\)

    27. ⓐ \(3·5^{−1}\) ⓑ \((3·5)^{−1}\)

    Answer

    ⓐ \(\dfrac{3}{5}\) ⓑ \(\dfrac{1}{15}\)

    28. ⓐ \(3·4^{−2}\) ⓑ \((3·4)^{−2}\)

    In the following exercises, simplify each expression using the Product Property.

    29. ⓐ \(b^{4}b^{−8}\) ⓑ \((w^{4}x^{−5})(w^{−2}x^{−4})\)) ⓒ \((−6c^{−3}d^9)(2c^4d^{−5})\)

    Answer

    ⓐ \(\dfrac{1}{b^{4}}\) ⓑ \(\dfrac{w^{2}}{x^{9}}\) ⓒ \(−12cd^{4}\)

    30. ⓐ \(s^{3}·s^{−7}\) ⓑ \((m^{3}n^{−3})(m^{5}n^{−1})\)
    ⓒ \((−2j^{−5}k^{8})(7j^{2}k^{−3})\)

    31. ⓐ \(a^{3}·a^{−3}\) ⓑ \((uv^{−2})(u^{−5}v^{−3})\)
    ⓒ \((−4r^{−2}s^{−8})(9r^{4}s^{3})\)

    Answer

    ⓐ \(1\) ⓑ \(\dfrac{1}{u^{4}v^{5}}\) ⓒ \(−36\dfrac{r^{2}}{j^{5}}\)

    32. ⓐ \(y^{5}·y^{−5}\) ⓑ \((pq^{−4})(p^{−6}q^{−3})\)
    ⓒ \((−5m^{4}n^{6})(8m^{−5}n^{−3})\)

    33. \(p^{5}·p^{−2}·p^{−4}\)

    Answer

    \(\dfrac{1}{p}\)

    34. \(x^{4}·x^{−2}·x^{−3}\)

    In the following exercises, simplify each expression using the Power Property.

    35. ⓐ \((m^4)^2\) ⓑ \((10^3)^6\) ⓒ \((x^3)^{−4}\)

    Answer

    ⓐ \(m^{8}\) ⓑ \(10^{18}\) ⓒ \(\dfrac{1}{x^{12}}\)

    36. ⓐ \((b^{2})^{7}\) ⓑ \((3^8)^2\) ⓒ \((k^2)^{−5}\)

    37. ⓐ \((y^3)^x\) ⓑ \((5^x)^x\) ⓒ \((q^6)^{−8}\)

    Answer

    ⓐ \(y^{3x}\) ⓑ \(5^{xy}\) ⓒ \(\dfrac{1}{q^{48}}\)

    38. ⓐ \((x^2)^y\) ⓑ \((7^a)^b\) ⓒ \((a^9)^{−10}\)

    In the following exercises, simplify each expression using the Product to a Power Property.

    39. ⓐ \((−3xy)^2\) ⓑ \((6a)^0\) ⓒ \((5x^2)^{−2}\) ⓓ \((−4y^{−3})^2\)

    Answer

    ⓐ \(9x^2y^2\) ⓑ 1 ⓒ \(\dfrac{1}{25x^4}\) ⓓ \(\dfrac{16}{y^6}\)

    40. ⓐ \((−4ab)^2\) ⓑ \((5x)^0\) ⓒ \((4y^3)^{−3}\) ⓓ \((−7y^{−3})^2\)

    41. ⓐ \((−5ab)^3\) ⓑ \((−4pq)^0\) ⓒ \((−6x^3)^{−2}\) ⓓ \((3y^{−4})^2\)

    Answer

    ⓐ \(−125a^3b^3\) ⓑ 1 ⓒ \(\dfrac{1}{36x^6}\) ⓓ \(\dfrac{9}{y^8}\)

    42. ⓐ \((−3xyz)^4\) ⓑ \((−7mn)^0\) ⓒ \((−3x^3)^{−2}\)
    ⓓ \((2y^{−5})^2\)

    In the following exercises, simplify each expression using the Quotient to a Power Property.

    43. ⓐ \((p^2)^5\) ⓑ \(\left(\dfrac{x}{y}\right)^{−6}\) ⓒ \(\left(\dfrac{2xy^2}{z}\right)^3\) ⓓ \(\left(\dfrac{4p^{−3}}{q^2}\right)^2\)

    Answer

    ⓐ \(\dfrac{p^5}{32}\) ⓑ \(\dfrac{y^6}{x^6}\) ⓒ \(\dfrac{8x^3y^6}{z^3}\)
    ⓓ \(\dfrac{16}{p^6q^4}\)

    44. ⓐ \(\left(\dfrac{x}{3}\right)^4\) ⓑ \(\left(\dfrac{a}{b}\right)^{−5}\) ⓒ \(\left(\dfrac{2xy^2}{z}\right)^3\) ⓓ \(\left(\dfrac{x^3y}{z^4}\right)^2\)

    45. ⓐ \(\left(\dfrac{a}{3b}\right)^4\) ⓑ \(\left(\dfrac{5}{4m}\right)^{−2}\) ⓒ \(\left(\dfrac{3a^{−2}b^3}{c^3}\right)^{−2}\) ⓓ \(\left(\dfrac{p^{−1}q^4}{r^{−4}}\right)^2\)

    Answer

    ⓐ \(\dfrac{a^4}{81b^4}\) ⓑ \(\dfrac{16m^2}{25}\) ⓒ \(\dfrac{a^4c^4}{9b^6}\) ⓓ \(\dfrac{q^8r^8}{p^2}\)

    46. ⓐ \(\left(\dfrac{x^2}{y}\right)^3\) ⓑ \(\left(\dfrac{10}{3q}\right)^{−4}\) ⓒ \(\left(\dfrac{2x^3y^4}{3z^2}\right)^5\) ⓓ \(\left(\dfrac{5a^3b^{−1}}{2c^4}\right)^{−3}\)

    In the following exercises, simplify each expression by applying several properties.

    47. ⓐ \((5t^2)^3(3t)^2\) ⓑ \(\dfrac{(t^2)^5(t^{−4})^2}{(t^3)^7}\) ⓒ \(\left(\dfrac{2xy^2}{x^3y^{−2}}\right)^2\left(\dfrac{12xy^3}{x^3y^{−1}}\right)^{−1}\)

    Answer

    ⓐ \(1125t^8\) ⓑ \(\dfrac{1}{t^{19}}\) ⓒ \(\dfrac{y^4}{3x^2}\)

    48. ⓐ \((10k^4)^3(5k^6)^2\) ⓑ \(\dfrac{(q^3)^6(q^{−2})^3}{(q^4)^8}\)

    49. ⓐ \((m^2n)^2(2mn^5)^4\) ⓑ \(\dfrac{(−2p^{−2})^4(3p^4)^2}{(−6p^3)^2}\)

    Answer

    ⓐ \(16m^8n^{22}\) ⓑ \(\dfrac{4}{p^6}\)

    50. ⓐ \((3pq^4)^2(6p^6q)^2\) ⓑ \(\dfrac{(−2k^{−3})^2(6k^2)^4}{(9k^4)^2}\)

    Mixed Practice

    In the following exercises, simplify each expression.

    51. ⓐ \(7n^{−1}\) ⓑ \((7n)^{−1}\) ⓒ \((−7n)^{−1}\)

    Answer

    ⓐ \(\dfrac{7}{n}\) ⓑ \(\dfrac{1}{7n}\) ⓒ \(−\dfrac{1}{7n}\)

    52. ⓐ \(6r^{−1}\) ⓑ \((6r)^{−1}\) ⓒ \((−6r)^{−1}\)

    53. ⓐ \((3p)^{−2}\) ⓑ \(3p^{−2}\) ⓒ \(−3p^{−2}\)

    Answer

    ⓐ \(\dfrac{1}{9p^2}\) ⓑ \(\dfrac{3}{p^2}\) ⓒ \(−\dfrac{3}{p^2}\)

    54. ⓐ \((2q)^{−4}\) ⓑ \(2q^{−4}\) ⓒ \(−2q^{−4}\)

    55. \((x^2)^4·(x^3)^2\)

    Answer

    \(x^{14}\)

    56. \((y^4)^3·(y^5)^2\)

    57. \((a^2)^6·(a^3)^8\)

    Answer

    \(a^{30}\)

    58. \((b^7)^5·(b^2)^6\)

    59. \((2m^6)^3\)

    Answer

    \(2m^{18}\)

    60. \((3y^2)^4\)

    61. \((10x^2y)^3\)

    Answer

    \(1,000x^6y^3\)

    62. \((2mn^4)^5\)

    63. \((−2a^3b^2)^4\)

    Answer

    \(16a^{12}b^8\)

    64. \((−10u^2v^4)^3\)

    65. \(\left(\dfrac{2}{3}x^2y\right)^3\)

    Answer

    \(\dfrac{8}{27}x^6y^3\)

    66. \(\left(\dfrac{7}{9}pq^4\right)^2\)

    67. \((8a^3)^2(2a)^4\)

    Answer

    \(1,024a^{10}\)

    68. \((5r^2)^3(3r)^2\)

    69. \((10p^4)^3(5p^6)^2\)

    Answer

    \(25,000p^{24}\)

    70. \((4x^3)^3(2x^5)^4\)

    71. \(\left(\dfrac{1}{2}x^2y^3\right)^4\left(4x^5y^3\right)^2\)

    Answer

    \(x^{18}y^{18}\)

    72. \(\left(\dfrac{1}{3}m^3n^2\right)^4\left(9m^8n^3\right)^2\)

    73. \((3m^2n)^2(2mn^5)^4\)

    Answer

    \(144m^8n^{22}\)

    74. \((2pq^4)^3(5p^6q)^2\)

    75. ⓐ \((3x)^2(5x)\) ⓑ \((2y)3(6y)\)

    Answer

    ⓐ \(45x^3\) ⓑ \(48y^4\)

    76. ⓐ \(\left(\dfrac{1}{2}y^2\right)^3\left(\dfrac{2}{3}y\right)^2\) ⓑ \(\left(\dfrac{1}{2}j^2\right)^5\left(\dfrac{2}{5}j^3\right)^2\)

    77. ⓐ \((2r^{−2})^3(4^{−1}r)^2\) ⓑ \((3x^{−3})^3(3^{−1}x^5)^4\)

    Answer

    ⓐ \(12r^4\) ⓑ \(13x^{11}\)

    78. \(\left(\dfrac{k^{−2}k^8}{k^3}\right)^2\)

    79. \(\left(\dfrac{j^{−2}j^5}{j^4}\right)^3\)

    Answer

    \(\dfrac{1}{j^3}\)

    80. \(\dfrac{(−4m^{−3})^2(5m^4)^3}{(−10m^6)^3}\)

    81. \(\dfrac{(−10n^{−2})^3(4n^5)^2}{(2n^8)^2}\)

    Answer

    \(−\dfrac{4000}{n^{12}}\)

    Use Scientific Notation

    In the following exercises, write each number in scientific notation.

    82. ⓐ 57,000 ⓑ 0.026

    83. ⓐ 340,000 ⓑ 0.041

    Answer

    ⓐ \(34\times10^4\) ⓑ \(41\times10^{−3}\)

    84. ⓐ 8,750,000 ⓑ 0.00000871

    85. ⓐ 1,290,000 ⓑ 0.00000103

    Answer

    ⓐ \(1.29\times10^6\)

    ⓑ \(103\times10^{−8}\)

    In the following exercises, convert each number to decimal form.

    86. ⓐ \(5.2\times10^2\) ⓑ \(2.5\times10^{−2}\)

    87. ⓐ \(−8.3\times10^2\) ⓑ \(3.8\times10^{−2}\)

    Answer

    ⓐ \(−830\) ⓑ 0.038

    88. ⓐ \(7.5\times10^6\) ⓑ \(−4.13\times10^{−5}\)

    89. ⓐ \(1.6\times10^{10}\) ⓑ \(8.43\times10^{−6}\)

    Answer

    ⓐ 16,000,000,000
    ⓑ 0.00000843

    In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

    90. ⓐ \((3\times10^{−5})(3\times10^9)\) ⓑ \(\dfrac{7\times10^{−3}}{1\times10^{−7}}\)

    91. ⓐ \((2\times10^2)(1\times10^{−4})\) ⓑ \(\dfrac{5\times10^{−2}}{1\times10^{−10}}\)

    Answer

    ⓐ 0.02 ⓑ 500,000,000

    92. ⓐ \((7.1\times10^{−2})(2.4\times10^{−4})\) ⓑ \(\dfrac{6\times10^4}{3\times10^{−2}}\)

    93. ⓐ \((3.5\times10^{−4})(1.6\times10^{−2})\) ⓑ \(\dfrac{8\times10^6}{4\times10^{−1}}\)

    Answer

    ⓐ 0.0000056 ⓑ 20,000,000

    Writing Exercises

    94. Use the Product Property for Exponents to explain why \(x·x=x^2\).

    95. Jennifer thinks the quotient \(\dfrac{a^{24}}{a^6}\) simplifies to \(a^4\). What is wrong with her reasoning?

    Answer

    Answers will vary.

    96. Explain why \(−5^3=(−5)^3\) but \(−5^4 \neq (−5)^4\).

    97. When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

    Answer

    Answers will vary.

    Self Check

    ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “simplify expressions using the properties for exponents.”, “use the definition of a negative exponent”, and “use scientific notation”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

    ⓑ After reviewing this checklist, what will you do to become confident for all goals?

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