Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

1.6E: Exercises

  • Page ID
    30292
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    Practice Makes Perfect

    Use the Commutative and Associative Properties

    In the following exercises, simplify.

    1.   \(43m+(−12n)+(−16m)+(−9n)\)

    Answer

    \(27m+(−21n)\)

    2.   \(−22p+17q+(−35p)+(−27q)\)

    3.   \(\frac{3}{8}g+\frac{1}{12}h+\frac{7}{8}g+\frac{5}{12}h\)

    Answer

    \(\frac{5}{4}g+\frac{1}{2}h\)

    4.   \(\frac{5}{6}a+\frac{3}{10}b+\frac{1}{6}a+\frac{9}{10}b\)

    5.   \(6.8p+9.14q+(−4.37p)+(−0.88q)\)

    Answer

    \(2.43p+8.26q\)

    6.   \(9.6m+7.22n+(−2.19m)+(−0.65n)\)

    7.   \(−24·7·\frac{3}{8}\)

    Answer

    \(−63\)

    8.   \(−36·11·\frac{4}{9}\)

    9.   \(\left(\frac{5}{6}+\frac{8}{15}\right)+\frac{7}{15}\)

    Answer

    \(1\frac{5}{6}\)

    10.   \(\left(\frac{11}{12}+\frac{4}{9}\right)+\frac{5}{9}\)

    11.   \(17(0.25)(4)\)

    Answer

    \(17\)

    12.   \(36(0.2)(5)\)

    13.   \([2.48(12)](0.5)\)

    Answer

    \(14.88\)

    14.   \([9.731(4)](0.75)\)

    15.   \(12\left(\frac{5}{6}p\right)\)

    Answer

    \(10p\)

    16.   \(20\left(\frac{3}{5}q\right)\)

    Use the Properties of Identity, Inverse and Zero

    In the following exercises, simplify.

    17.   \(19a+44−19a\)

    Answer

    \(44\)

    18.   \(27c+16−27c\)

    19.   \(\frac{1}{2}+\frac{7}{8}+\left(−\frac{1}{2}\right)\)

    Answer

    \(\frac{7}{8}\)

    20.   \(\frac{2}{5}+\frac{5}{12}+\left(−\frac{2}{5}\right)\)

    21.   \(10(0.1d)\)

    Answer

    \(d\)

    22.   \(100(0.01p)\)

    23. \(\frac{3}{20}·\frac{49}{11}·\frac{20}{3}\)

    Answer

    \(\frac{49}{11}\)

    24.   \(\frac{13}{18}·\frac{25}{7}·\frac{18}{13}\)

    25.   \(\frac{0}{u−4.99}\), where \(u\neq 4.99\)

    Answer

    \(0\)

    26.   \(0÷(y−\frac{1}{6})\), where \(x \neq 16\)

    27.   \(\frac{32−5a}{0}\), where \(32−5a\neq 0\)

     
    Answer

    undefined

    28.   \(\frac{28−9b}{0}\), where \(28−9b\neq 0\)

    29.   \(\left(\frac{3}{4}+\frac{9}{10}m\right)÷0\), where \(\frac{3}{4}+\frac{9}{10}m\neq 0\)

    Answer

    undefined

    30.   \(\left(\frac{5}{16}n−\frac{3}{7}\right)÷0\), where \(\frac{5}{16}n−\frac{3}{7}\neq 0\)

     

    Simplify Expressions Using the Distributive Property

    In the following exercises, simplify using the Distributive Property.

    31.   \(8(4y+9)\)

    Answer

    \(32y+72\)

    32.   \(9(3w+7)\)

    33.   \(6(c−13)\)

    Answer

    \(6c−78\)

    34.   \(7(y−13)\)

    35.   \(\frac{1}{4}(3q+12)\)

    Answer

    \(\frac{3}{4}q+3\)

    36.   \(\frac{1}{5}(4m+20)\)

    37.   \(9(\frac{5}{9}y−\frac{1}{3})\)

    Answer

    \(5y−3\)

    38.   \(10(\frac{3}{10}x−\frac{2}{5})\)

    39.   \(12(\frac{1}{4}+\frac{2}{3}r)\)

    Answer

    \(3+8r\)

    40.   \(12(\frac{1}{6}+\frac{3}{4}s)\)

    41.   \(15⋅\frac{3}{5}(4d+10)\)

    Answer

    \(36d+90\)

    42.   \(18⋅\frac{5}{6}(15h+24)\)

    43.   \(r(s−18)\)

    Answer

    \(rs−18r\)

    44.   \(u(v−10)\)

    45.   \((y+4)p\)

    Answer

    \(yp+4p\)

    46.   \((a+7)x\)

    47.   \(−7(4p+1)\)

    Answer

    \(−28p−7\)

    48.   \(−9(9a+4)\)

    49.   \(−3(x−6)\)

    Answer

    \(−3x+18\)

    50.   \(−4(q−7)\)

    51.   \(−(3x−7)\)

    Answer

    \(−3x+7\)

    52.   \(−(5p−4)\)

    53.   \(16−3(y+8)\)

    Answer

    \(−3y−8\)

    54.   \(18−4(x+2)\)

    55.   \(4−11(3c−2)\)

    Answer

    \(−33c+26\)

    56.   \(9−6(7n−5)\)

    57.   \(22−(a+3)\)

    Answer

    \(−a+19\)

    58.   \(8−(r−7)\)

    59.   \((5m−3)−(m+7)\)

    Answer

    \(4m−10\)

    60.   \((4y−1)−(y−2)\)

    61.   \(9(8x−3)−(−2)\)

    Answer

    \(72x−25\)

    62.   \(4(6x−1)−(−8)\)

    63.   \(5(2n+9)+12(n−3)\)

    Answer

    \(22n+9\)

    64.   \(9(5u+8)+2(u−6)\)

    65.   \(14(c−1)−8(c−6)\)

    Answer

    \(6c+34\)

    66.   \(11(n−7)−5(n−1)\)

    67.   \(6(7y+8)−(30y−15)\)

    Answer

    \(12y+63\)

    68.   \(7(3n+9)−(4n−13)\)

    Writing Exercises

    69.  In your own words, state the Associative Property of addition.

    Answer

    Answers will vary.

    70.  What is the difference between the additive inverse and the multiplicative inverse of a number

    71.  Simplify \(8(x−\frac{1}{4})\) using the Distributive Property and explain each step.

    Answer

    Answers will vary.

    72.  Explain how you can multiply \(4($5.97)\) without paper or calculator by thinking of \($5.97\) as \(6−0.03\) and then using the Distributive Property.

    Self Check

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

    This table has 4 columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first column has the following statements: use the commutative and associative properties, use the properties of identity, inverse and zero, simplify expressions using the Distributive Property. The remaining columns are blank.

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