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Mathematics LibreTexts

6.5E: Exercises

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    30429
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    Practice Makes Perfect

    Simplify Expressions Using the Quotient Property for Exponents

    In the following exercises, simplify.

    Exercise \(\PageIndex{1}\)

    1. \(\dfrac{x^{18}}{x^{3}}\)
    2. \(\dfrac{5^{12}}{5^{3}}\)

    Exercise \(\PageIndex{2}\)

    1. \(\dfrac{y^{20}}{y^{10}}\)
    2. \(\dfrac{7^{16}}{7^{2}}\)
    Answer
    1. \(y^{10}\)
    2. \(7^{14}\)

    Exercise \(\PageIndex{3}\)

    1. \(\dfrac{p^{21}}{p^{7}}\)
    2. \(\dfrac{4^{16}}{4^{4}}\)

    Exercise \(\PageIndex{4}\)

    1. \(\dfrac{u^{24}}{u^{3}}\)
    2. \(\dfrac{9^{15}}{9^{5}}\)
    Answer
    1. \(u^{21}\)
    2. \(9^{10}\)

    Exercise \(\PageIndex{5}\)

    1. \(\dfrac{q^{18}}{q^{36}}\)
    2. \(\dfrac{10^{2}}{10^{3}}\)

    Exercise \(\PageIndex{6}\)

    1. \(\dfrac{t^{10}}{t^{40}}\)
    2. \(\dfrac{8^{3}}{8^{5}}\)
    Answer
    1. \(\dfrac{1}{t^{30}}\)
    2. \(\dfrac{1}{64}\)

    Exercise \(\PageIndex{7}\)

    1. \(\dfrac{b}{b^{9}}\)
    2. \(\dfrac{4}{4^{6}}\)

    Exercise \(\PageIndex{8}\)

    1. \(\dfrac{x}{x^{7}}\)
    2. \(\dfrac{10}{10^{3}}\)
    Answer
    1. \(\dfrac{1}{x^{6}}\)
    2. \(\dfrac{1}{100}\)

    Simplify Expressions with Zero Exponents

    In the following exercises, simplify.

    Exercise \(\PageIndex{9}\)

    1. \(20^{0}\)
    2. \(b^{0}\)

    Exercise \(\PageIndex{10}\)

    1. \(13^0\)
    2. \(k^{0}\)
    Answer
    1. 1
    2. 1

    Exercise \(\PageIndex{11}\)

    1. \(-27^{0}\)
    2. \(-\left(27^{0}\right)\)

    Exercise \(\PageIndex{12}\)

    1. \(-15^{0}\)
    2. \(-\left(15^{0}\right)\)
    Answer
    1. −1
    2. −1

    Exercise \(\PageIndex{13}\)

    1. \((25 x)^{0}\)
    2. \(25 x^{0}\)

    Exercise \(\PageIndex{14}\)

    1. \((6 y)^{0}\)
    2. \(6 y^{0}\)
    Answer
    1. 1
    2. 6

    Exercise \(\PageIndex{15}\)

    1. \((12 x)^{0}\)
    2. \(\left(-56 p^{4} q^{3}\right)^{0}\)

    Exercise \(\PageIndex{16}\)

    1. 7\(y^{0}(17 y)^{0}\)
    2. \(\left(-93 c^{7} d^{15}\right)^{0}\)
    Answer
    1. 7
    2. 1

    Exercise \(\PageIndex{17}\)

    1. \(12 n^{0}-18 m^{0}\)
    2. \((12 n)^{0}-(18 m)^{0}\)

    Exercise \(\PageIndex{18}\)

    1. \(15 r^{0}-22 s^{0}\)
    2. \((15 r)^{0}-(22 s)^{0}\)
    Answer
    1. −7
    2. 0

    Simplify Expressions Using the Quotient to a Power Property

    In the following exercises, simplify.

    Exercise \(\PageIndex{19}\)

    1. \(\left(\dfrac{3}{4}\right)^{3}\)
    2. \(\left(\dfrac{p}{2}\right)^{5}\)
    3. \(\left(\dfrac{x}{y}\right)^{6}\)

    Exercise \(\PageIndex{20}\)

    1. \(\left(\dfrac{2}{5}\right)^{2}\)
    2. \(\left(\dfrac{x}{3}\right)^{4}\)
    3. \(\left(\dfrac{a}{b}\right)^{5}\)
    Answer
    1. \(\dfrac{4}{25}\)
    2. \(\dfrac{x^{4}}{81}\)
    3. \(\left(\dfrac{a}{b}\right)^{5}\)

    Exercise \(\PageIndex{21}\)

    1. \(\left(\dfrac{a}{3 b}\right)^{4}\)
    2. \(\left(\dfrac{5}{4 m}\right)^{2}\)

    Exercise \(\PageIndex{22}\)

    1. \(\left(\dfrac{a}{3 b}\right)^{4}\)
    2. \(\left(\dfrac{10}{3 q}\right)^{4}\)
    Answer
    1. \(\dfrac{x^{3}}{8 y^{3}}\)
    2. \(\dfrac{10,000}{81 q^{4}}\)

    Simplify Expressions by Applying Several Properties

    In the following exercises, simplify.

    Exercise \(\PageIndex{23}\)

    \(\dfrac{\left(a^{2}\right)^{3}}{a^{4}}\)

    Exercise \(\PageIndex{24}\)

    \(\dfrac{\left(p^{3}\right)^{4}}{p^{5}}\)

    Answer

    \(p^{7}\)

    Exercise \(\PageIndex{25}\)

    \(\dfrac{\left(y^{3}\right)^{4}}{y^{10}}\)

    Exercise \(\PageIndex{26}\)

    \(\dfrac{\left(x^{4}\right)^{5}}{x^{15}}\)

    Answer

    \(x^{5}\)

    Exercise \(\PageIndex{27}\)

    \(\dfrac{u^{6}}{\left(u^{3}\right)^{2}}\)

    Exercise \(\PageIndex{28}\)

    \(\dfrac{v^{20}}{\left(v^{4}\right)^{5}}\)

    Answer

    1

    Exercise \(\PageIndex{29}\)

    \(\dfrac{m^{12}}{\left(m^{8}\right)^{3}}\)

    Exercise \(\PageIndex{30}\)

    \(\dfrac{n^{8}}{\left(n^{6}\right)^{4}}\)

    Answer

    \(\dfrac{1}{n^{16}}\)

    Exercise \(\PageIndex{31}\)

    \(\left(\dfrac{p^{9}}{p^{3}}\right)^{5}\)

    Exercise \(\PageIndex{32}\)

    \(\left(\dfrac{q^{8}}{q^{2}}\right)^{3}\)

    Answer

    \(q^{18}\)

    Exercise \(\PageIndex{33}\)

    \(\left(\dfrac{r^{2}}{r^{6}}\right)^{3}\)

    Exercise \(\PageIndex{34}\)

    \(\left(\dfrac{m^{4}}{m^{7}}\right)^{4}\)

    Answer

    \(\dfrac{1}{m^{12}}\)

    Exercise \(\PageIndex{35}\)

    \(\left(\dfrac{p}{r^{11}}\right)^{2}\)

    Exercise \(\PageIndex{36}\)

    \(\left(\dfrac{a}{b^{6}}\right)^{3}\)

    Answer

    \(\dfrac{a^{3}}{b^{18}}\)

    Exercise \(\PageIndex{37}\)

    \(\left(\dfrac{w^{5}}{x^{3}}\right)^{8}\)

    Exercise \(\PageIndex{38}\)

    \(\left(\dfrac{y^{4}}{z^{10}}\right)^{5}\)

    Answer

    \(\dfrac{y^{20}}{z^{50}}\)

    Exercise \(\PageIndex{39}\)

    \(\left(\dfrac{2 j^{3}}{3 k}\right)^{4}\)

    Exercise \(\PageIndex{40}\)

    \(\left(\dfrac{3 m^{5}}{5 n}\right)^{3}\)

    Answer

    \(\dfrac{27 m^{15}}{125 n^{3}}\)

    Exercise \(\PageIndex{41}\)

    \(\left(\dfrac{3 c^{2}}{4 d^{6}}\right)^{3}\)

    Exercise \(\PageIndex{42}\)

    \(\left(\dfrac{5 u^{7}}{2 v^{3}}\right)^{4}\)

    Answer

    \(\dfrac{625 u^{28}}{16 v^{12}}\)

    Exercise \(\PageIndex{43}\)

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

    Exercise \(\PageIndex{44}\)

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

    Answer

    \(j^{9}\)

    Exercise \(\PageIndex{45}\)

    \(\dfrac{\left(t^{2}\right)^{5}\left(t^{4}\right)^{2}}{\left(t^{3}\right)^{7}}\)

    Exercise \(\PageIndex{46}\)

    \(\dfrac{\left(q^{3}\right)^{6}\left(q^{2}\right)^{3}}{\left(q^{4}\right)^{8}}\)

    Answer

    \(\dfrac{1}{q^{8}}\)

    Exercise \(\PageIndex{47}\)

    \(\dfrac{\left(-2 p^{2}\right)^{4}\left(3 p^{4}\right)^{2}}{\left(-6 p^{3}\right)^{2}}\)

    Exercise \(\PageIndex{48}\)

    \(\dfrac{\left(-2 k^{3}\right)^{2}\left(6 k^{2}\right)^{4}}{\left(9 k^{4}\right)^{2}}\)

    Answer

    64\(k^{6}\)

    Exercise \(\PageIndex{49}\)

    \(\dfrac{\left(-4 m^{3}\right)^{2}\left(5 m^{4}\right)^{3}}{\left(-10 m^{6}\right)^{3}}\)

    Exercise \(\PageIndex{50}\)

    \(\dfrac{\left(-10 n^{2}\right)^{3}\left(4 n^{5}\right)^{2}}{\left(2 n^{8}\right)^{2}}\)

    Answer

    −4,000

    Divide Monomials

    In the following exercises, divide the monomials.

    Exercise \(\PageIndex{51}\)

    56\(b^{8} \div 7 b^{2}\)

    Exercise \(\PageIndex{52}\)

    63\(\nu^{10} \div 9 v^{2}\)

    Answer

    7\(v^{8}\)

    Exercise \(\PageIndex{53}\)

    \(-88 y^{15} \div 8 y^{3}\)

    Exercise \(\PageIndex{54}\)

    \(-72 u^{12} \div 12 u^{4}\)

    Answer

    \(-6 u^{8}\)

    Exercise \(\PageIndex{55}\)

    \(\dfrac{45 a^{6} b^{8}}{-15 a^{10} b^{2}}\)

    Exercise \(\PageIndex{56}\)

    \(\dfrac{54 x^{9} y^{3}}{-18 x^{6} y^{15}}\)

    Answer

    \(-\dfrac{3 x^{3}}{y^{12}}\)

    Exercise \(\PageIndex{57}\)

    \(\dfrac{15 r^{4} s^{9}}{18 r^{9} s^{2}}\)

    Exercise \(\PageIndex{58}\)

    \(\dfrac{20 m^{8} n^{4}}{30 m^{5} n^{9}}\)

    Answer

    \(\dfrac{-2 m^{3}}{3 n^{5}}\)

    Exercise \(\PageIndex{59}\)

    \(\dfrac{18 a^{4} b^{8}}{-27 a^{9} b^{5}}\)

    Exercise \(\PageIndex{60}\)

    \(\dfrac{45 x^{5} y^{9}}{-60 x^{8} y^{6}}\)

    Answer

    \(\dfrac{-3 y^{3}}{4 x^{3}}\)

    Exercise \(\PageIndex{61}\)

    \(\dfrac{64 q^{11} r^{9} s^{3}}{48 q^{6} r^{8} s^{5}}\)

    Exercise \(\PageIndex{62}\)

    \(\dfrac{65 a^{10} b^{8} c^{5}}{42 a^{7} b^{6} c^{8}}\)

    Answer

    \(\dfrac{65 a^{3} b^{2}}{42 c^{3}}\)

    Exercise \(\PageIndex{63}\)

    \(\dfrac{\left(10 m^{5} n^{4}\right)\left(5 m^{3} n^{6}\right)}{25 m^{7} n^{5}}\)

    Exercise \(\PageIndex{64}\)

    \(\dfrac{\left(-18 p^{4} q^{7}\right)\left(-6 p^{3} q^{8}\right)}{-36 p^{12} q^{10}}\)

    Answer

    \(\dfrac{-3 q^{5}}{p^{5}}\)

    Exercise \(\PageIndex{65}\)

    \(\dfrac{\left(6 a^{4} b^{3}\right)\left(4 a b^{5}\right)}{\left(12 a^{2} b\right)\left(a^{3} b\right)}\)

    Exercise \(\PageIndex{66}\)

    \(\dfrac{\left(4 u^{2} v^{5}\right)\left(15 u^{3} v\right)}{\left(12 u^{3} v\right)\left(u^{4} v\right)}\)

    Answer

    \(\dfrac{5 v^{4}}{u^{2}}\)

    Mixed Practice

    Exercise \(\PageIndex{67}\)

    1. \(24 a^{5}+2 a^{5}\)
    2. \(24 a^{5}-2 a^{5}\)
    3. 24\(a^{5} \cdot 2 a^{5}\)
    4. 24\(a^{5} \div 2 a^{5}\)

    Exercise \(\PageIndex{68}\)

    1. \(15 n^{10}+3 n^{10}\)
    2. \(15 n^{10}-3 n^{10}\)
    3. 15\(n^{10} \cdot 3 n^{10}\)
    4. 15\(n^{10} \div 3 n^{10}\)
    Answer
    1. 18\(n^{10}\)
    2. 12\(n^{10}\)
    3. 45\(n^{20}\)
    4. 5

    Exercise \(\PageIndex{69}\)

    1. \(p^{4} \cdot p^{6}\)
    2. \(\left(p^{4}\right)^{6}\)

    Exercise \(\PageIndex{70}\)

    1. \(q^{5} \cdot q^{3}\)
    2. \(\left(q^{5}\right)^{3}\)
    Answer
    1. \(q^{8}\)
    2. \(q^{15}\)

    Exercise \(\PageIndex{71}\)

    1. \(\dfrac{y^{3}}{y}\)
    2. \(\dfrac{y}{y^{3}}\)

    Exercise \(\PageIndex{72}\)

    1. \(\dfrac{z^{6}}{z^{5}}\)
    2. \(\dfrac{z^{5}}{z^{6}}\)
    Answer
    1. z
    2. \(\dfrac{1}{z}\)

    Exercise \(\PageIndex{73}\)

    \(\left(8 x^{5}\right)(9 x) \div 6 x^{3}\)

    Exercise \(\PageIndex{74}\)

    \((4 y)\left(12 y^{7}\right) \div 8 y^{2}\)

    Answer

    6\(y^{6}\)

    Exercise \(\PageIndex{75}\)

    \(\dfrac{27 a^{7}}{3 a^{3}}+\dfrac{54 a^{9}}{9 a^{5}}\)

    Exercise \(\PageIndex{76}\)

    \(\dfrac{32 c^{11}}{4 c^{5}}+\dfrac{42 c^{9}}{6 c^{3}}\)

    Answer

    15\(c^{6}\)

    Exercise \(\PageIndex{77}\)

    \(\dfrac{32 y^{5}}{8 y^{2}}-\dfrac{60 y^{10}}{5 y^{7}}\)

    Exercise \(\PageIndex{78}\)

    \(\dfrac{48 x^{6}}{6 x^{4}}-\dfrac{35 x^{9}}{7 x^{7}}\)

    Answer

    3\(x^{2}\)

    Exercise \(\PageIndex{79}\)

    \(\dfrac{63 r^{6} s^{3}}{9 r^{4} s^{2}}-\dfrac{72 r^{2} s^{2}}{6 s}\)

    Exercise \(\PageIndex{80}\)

    \(\dfrac{56 y^{4} z^{5}}{7 y^{3} z^{3}}-\dfrac{45 y^{2} z^{2}}{5 y}\)

    Answer

    \(-y z^{2}\)

    Everyday Math

    Exercise \(\PageIndex{81}\)

    Memory One megabyte is approximately \(10^6\) bytes. One gigabyte is approximately \(10^9\) bytes. How many megabytes are in one gigabyte?

    Exercise \(\PageIndex{82}\)

    Memory One gigabyte is approximately \(10^9\) bytes. One terabyte is approximately \(10^12\) bytes. How many gigabytes are in one terabyte?

    Answer

    \(10^{3}\)

    Writing Exercises

    Exercise \(\PageIndex{83}\)

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

    Exercise \(\PageIndex{84}\)

    Maurice simplifies the quotient \(\dfrac{d^{7}}{d}\) by writing \(\dfrac{\not{d}^7}{\not{d}}=7 .\) What is wrong with his reasoning?

    Answer

    Answers will vary.

    Exercise \(\PageIndex{85}\)

    When Drake simplified \(-3^{0}\) and \((-3)^{0}\) he got the same answer. Explain how using the Order of Operations correctly gives
    different answers.

    Exercise \(\PageIndex{86}\)

    Robert thinks \(x^{0}\) simplifies to 0. What would you say to convince Robert he is wrong?

    Answer

    Answers will vary.

    Self Check

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

    This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “simplify expressions using the Quotient Property for Exponents,” “simplify expressions with zero exponents,” “simplify expressions using the Quotient to a Power Property,” “simplify expressions by applying several properties,” and “divide monomials.” The rest of the cells are blank.

    b. On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?