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8.7E: Exercises

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    Practice Makes Perfect

    Simplify Expressions with Higher Roots

    In the following exercises, simplify.

    Example \(\PageIndex{46}\)
    1. \(\sqrt[3]{216}\)
    2. \(\sqrt[4]{256}\)
    3. \(\sqrt[5]{32}\)
    Example \(\PageIndex{47}\)
    1. \(\sqrt[3]{27}\)
    2. \(\sqrt[4]{16}\)
    3. \(\sqrt[5]{243}\)
    Answer
    1. 3
    2. 2
    3. 3
    Example \(\PageIndex{48}\)
    1. \(\sqrt[3]{512}\)
    2. \(\sqrt[4]{81}\)
    3. \(\sqrt[5]{1}\)
    Example \(\PageIndex{49}\)
    1. \(\sqrt[5]{125}\)
    2. \(\sqrt[4]{1296}\)
    3. \(\sqrt[5]{1024}\)
    Answer
    1. 5
    2. 6
    3. 4
    Example \(\PageIndex{50}\)
    1. \(\sqrt[3]{−8}\)
    2. \(\sqrt[4]{−81}\)
    3. \(\sqrt[5]{−32}\)
    Example \(\PageIndex{51}\)
    1. \(\sqrt[3]{−64}\)
    2. \(\sqrt[4]{−16}\)
    3. \(\sqrt[5]{−243}\)
    Answer
    1. −4
    2. not real
    3. −3
    Example \(\PageIndex{52}\)
    1. \(\sqrt[3]{−125}\)
    2. \(\sqrt[4]{−1296}\)
    3. \(\sqrt[5]{−1024}\)
    Example \(\PageIndex{53}\)
    1. \(\sqrt[3]{−512}\)
    2. \(\sqrt[4]{−81}\)
    3. \(\sqrt[5]{−1}\)
    Answer
    1. −8
    2. not a real number
    3. −1
    Example \(\PageIndex{54}\)
    1. \(\sqrt[5]{u^5}\)
    2. \(\sqrt[8]{v^8}\)
    Example \(\PageIndex{55}\)
    1. \(\sqrt[3]{a^3}\)

    .

    Answer
    1. a
    2. |b|
    Example \(\PageIndex{56}\)
    1. \(\sqrt[4]{y^4}\)
    2. \(\sqrt[7]{m^7}\)
    Example \(\PageIndex{57}\)
    1. \(\sqrt[8]{k^8}\)
    2. \(\sqrt[6]{p^6}\)
    Answer
    1. |k|
    2. ∣p∣
    Example \(\PageIndex{58}\)
    1. \(\sqrt[3]{x^9}\)
    2. \(\sqrt[4]{y^{12}}\)
    Example \(\PageIndex{59}\)
    1. \(\sqrt[5]{a^{10}}\)
    2. \(\sqrt[3]{b^{27}}\)
    Answer
    1. \(a^2\)
    2. \(b^9\)
    Example \(\PageIndex{60}\)
    1. \(\sqrt[4]{m^8}\)
    2. \(\sqrt[5]{n^{20}}\)
    Example \(\PageIndex{61}\)
    1. \(\sqrt[6]{r^{12}}\)
    2. \(\sqrt[3]{s^{30}}\)
    Answer
    1. \(r^2\)
    2. \(s^{10}\)
    Example \(\PageIndex{62}\)
    1. \(\sqrt[4]{16x^8}\)
    2. \(\sqrt[6]{64y^{12}}\)
    Example \(\PageIndex{63}\)
    1. \(\sqrt[3]{−8c^9}\)
    2. \(\sqrt[3]{125d^{15}}\)
    Answer
    1. \(−2c^3\)
    2. \(5d^5\)
    Example \(\PageIndex{64}\)
    1. \(\sqrt[3]{216a^6}\)
    2. \(\sqrt[5]{32b^{20}}\)
    Example \(\PageIndex{65}\)
    1. \(\sqrt[7]{128r^{14}}\)
    2. \(\sqrt[4]{81s^{24}}\)
    Answer
    1. \(2r^2\)
    2. \(3s^6\)

    Use the Product Property to Simplify Expressions with Higher Roots

    In the following exercises, simplify.

    Exercise \(\PageIndex{66}\)
    1. \(\sqrt[3]{r^5}\)
    2. \(\sqrt[4]{s^{10}}\)
    Example \(\PageIndex{67}\)
    1. \(\sqrt[5]{u^7}\)
    2. \(\sqrt[6]{v^{11}}\)
    Answer
    1. \(u\sqrt[5]{u^2}\)
    2. \(v\sqrt[6]{v^5}\)
    Example \(\PageIndex{68}\)
    1. \(\sqrt[4]{m^5}\)
    2. \(\sqrt[8]{n^{10}}\)
    Example \(\PageIndex{69}\)
    1. \(\sqrt[5]{p^8}\)
    2. \(\sqrt[3]{q^8}\)
    Answer
    1. \(p\sqrt[5]{p^3}\)
    2. \(q^2\sqrt[3]{q^2}\)
    Example \(\PageIndex{70}\)
    1. \(\sqrt[4]{32}\)
    2. \(\sqrt[5]{64}\)
    Example \(\PageIndex{71}\)
    1. \(\sqrt[3]{625}\)
    2. \(\sqrt[6]{128}\)
    Answer
    1. \(5\sqrt[3]{5}\)
    2. \(2\sqrt[6]{2}\)
    Example \(\PageIndex{72}\)
    1. \(\sqrt[6]{64}\)
    2. \(\sqrt[3]{256}\)
    Example \(\PageIndex{73}\)
    1. \(\sqrt[4]{3125}\)
    2. \(\sqrt[3]{81}\)
    Answer
    1. \(5\sqrt[4]{5}\)
    2. \(3\sqrt[3]{3}\)
    Example \(\PageIndex{74}\)
    1. \(\sqrt[3]{108x^5}\)
    2. \(\sqrt[4]{48y^6}\)
    Example \(\PageIndex{75}\)
    1. \(\sqrt[5]{96a^7}\)
    2. \(\sqrt[3]{375b^4}\)
    Answer
    1. \(2a\sqrt[5]{3a^2}\)
    2. \(5b\sqrt[3]{3b}\)
    Example \(\PageIndex{76}\)
    1. \(\sqrt[4]{405m^{10}}\)
    2. \(\sqrt[5]{160n^8}\)
    Example \(\PageIndex{77}\)
    1. \(\sqrt[3]{512p^5}\)
    2. \(\sqrt[4]{324q^7}\)
    Answer
    1. \(8p\sqrt[3]{p^2}\)
    2. \(3q\sqrt[4]{4q^3}\)
    Example \(\PageIndex{78}\)
    1. \(\sqrt[3]{−864}\)
    2. \(\sqrt[4]{−256}\)
    Example \(\PageIndex{79}\)
    1. \(\sqrt[5]{−486}\)
    2. \(\sqrt[6]{−64}\)
    Answer
    1. \(−3\sqrt[5]{2}\)
    2. not real
    Example \(\PageIndex{80}\)
    1. \(\sqrt[5]{−32}\)
    2. \(\sqrt[8]{−1}\)
    Example \(\PageIndex{81}\)
    1. \(\sqrt[3]{−8}\)
    2. \(\sqrt[4]{−16}\)
    Answer
    1. −2
    2. not real
    ​​​​​​Use the Quotient Property to Simplify Expressions with Higher Roots

    In the following exercises, simplify.

    Example \(\PageIndex{82}\)
    1. \(\sqrt[3]{\frac{p^{11}}{p^2}}\)
    2. \(\sqrt[4]{\frac{q^{17}}{q^{13}}}\)
    Example \(\PageIndex{83}\)
    1. \(\sqrt[5]{\frac{d^{12}}{d^7}}\)
    2. \(\sqrt[8]{\frac{m^{12}}{m^4}}\)
    Answer
    1. d
    2. |m|
    Example \(\PageIndex{84}\)
    1. \(\sqrt[5]{\frac{u^{21}}{u^{11}}}\)
    2. \(\sqrt[6]{\frac{v^{30}}{v^{12}}}\)
    Example \(\PageIndex{85}\)
    1. \(\sqrt[3]{\frac{r^{14}}{r^5}}\)
    2. \(\sqrt[4]{\frac{c^{21}}{c^9}}\)
    Answer
    1. \(r^2\)
    2. \(∣c^3∣\)
    Example \(\PageIndex{86}\)
    1. \(\frac{\sqrt[4]{64}}{\sqrt[4]{2}}\)
    2. \(\frac{\sqrt[5]{128x^8}}{\sqrt[5]{2x^2}}\)
    Example \(\PageIndex{87}\)
    1. \(\frac{\sqrt[3]{−625}}{\sqrt[3]{5}}\)
    2. \(\frac{\sqrt[4]{80m^7}}{\sqrt[4]{5m}}\)
    Answer
    1. −5
    2. \(4m\sqrt[4]{m^2}\)
    Example \(\PageIndex{88}\)
    1. \(\sqrt[3]{\frac{1050}{2}}\)
    2. \(\sqrt[4]{\frac{486y^9}{2y^3}}\)
    Example \(\PageIndex{89}\)
    1. \(\sqrt[3]{\frac{162}{6}}\)
    2. \(\sqrt[4]{\frac{160r^{10}}{5r^3}}\)
    Answer
    1. \(3\sqrt[3]{6}\)
    2. \(2|r|\sqrt[4]{2r^3}\)
    Example \(\PageIndex{90}\)
    1. \(\sqrt[3]{\frac{54a^8}{b^3}}\)
    2. \(\sqrt[4]{\frac{64c^5}{d^2}}\)
    Example \(\PageIndex{91}\)
    1. \(\sqrt[5]{\frac{96r^{11}}{s^{3}}}\)
    2. \(\sqrt[6]{\frac{128u^7}{v^3}}\)
    Answer
    1. \(\frac{2r^2\sqrt[5]{3r}}{s^3}\)
    2. \(\frac{2u\sqrt[6]{2uv^3}}{v}\)
    Example \(\PageIndex{92}\)
    1. \(\sqrt[3]{\frac{81s^8}{t^3}}\)
    2. \(\sqrt[4]{\frac{64p^{15}}{q^{12}}}\)
    Example \(\PageIndex{93}\)
    1. \(\sqrt[3]{\frac{625u^{10}}{v^3}}\)
    2. \(\sqrt[4]{\frac{729c^{21}}{d^8}}\)
    Answer
    1. \(\frac{5u^3\sqrt[3]{5u}}{v}\)
    2. \(\frac{3c^5\sqrt[4]{9c}}{d^2}\)
    Add and Subtract Higher Roots

    In the following exercises, simplify.

    Example \(\PageIndex{94}\)
    1. \(\sqrt[7]{8p}+\sqrt[7]{8p}\)
    2. \(3\sqrt[3]{25}−\sqrt[3]{25}\)
    Example \(\PageIndex{95}\)
    1. \(\sqrt[3]{15q}+\sqrt[3]{15q}\)
    2. \(2\sqrt[4]{27}−6\sqrt[4]{27}\)
    Answer
    1. \(2\sqrt[3]{15q}\)
    2. \(−4\sqrt[4]{27}\)
    Example \(\PageIndex{96}\)
    1. \(3\sqrt[5]{9x}+7\sqrt[5]{9x}\)
    2. \(8\sqrt[7]{3q}−2\sqrt[7]{3q}\)
    Example \(\PageIndex{97}\)

    1.

    .

    2.

    .

    Answer

    1.

    .

    2.

    .

    Example \(\PageIndex{98}\)
    1. \(\sqrt[3]{81}−\sqrt[3]{192}\)
    2. \(\sqrt[4]{512}−\sqrt[4]{32}\)​​​​​​​
    Example \(\PageIndex{99}\)
    1. \(\sqrt[3]{250}−\sqrt[3]{54}\)
    2. \(\sqrt[4]{243}−\sqrt[4]{1875}\)
    Answer
    1. \(5\sqrt[3]{5}−3\sqrt[3]{2}\)
    2. \(−2\sqrt[4]{3}\)
    Example \(\PageIndex{100}\)
    1. \(\sqrt[3]{128}+\sqrt[3]{250}\)
    2. \(\sqrt[5]{729}+\sqrt[5]{96}\)
    Example \(\PageIndex{101}\)
    1. \(\sqrt[4]{243}+\sqrt[4]{1250}\)
    2. \(\sqrt[3]{2000}+\sqrt[3]{54}\)
    Answer
    1. \(3\sqrt[4]{3}+5\sqrt[4]{2}\)
    2. \(13\sqrt[3]{2}\)
    Example \(\PageIndex{102}\)
    1. \(\sqrt[3]{64a^{10}}−\sqrt[3]{−216a^{12}}\)
    2. \(\sqrt[4]{486u^7}+\sqrt[4]{768u^3}\)
    Example \(\PageIndex{103}\)
    1. \(\sqrt[3]{80b^5}−\sqrt[3]{−270b^3}\)
    2. \(\sqrt[4]{160v^{10}}−\sqrt[4]{1280v^3}\)
    Answer
    1. \(2b\sqrt[3]{10b^2}+3b\sqrt[3]{10}\)
    2. \(2v^2\sqrt[4]{10v^2}−4\sqrt[4]{5v^3}\)
    ​​​​​​Mixed Practice

    In the following exercises, simplify.

    Example \(\PageIndex{104}\)

    \(\sqrt[4]{16}\)

    Example \(\PageIndex{105}\)

    \(\sqrt[6]{64}\)

    Answer

    2

    Example \(\PageIndex{106}\)

    \(\sqrt[3]{a^3}\)

    Example \(\PageIndex{107}\)

    .

    Answer

    |b|

    Example \(\PageIndex{108}\)

    \(\sqrt[3]{−8c^9}\)

    Example \(\PageIndex{109}\)

    \(\sqrt[3]{125d^{15}}\)

    Answer

    \(5d^5\)

    Example \(\PageIndex{110}\)

    \(\sqrt[3]{r^5}\)

    Example \(\PageIndex{111}\)

    \(\sqrt[4]{s^{10}}\)

    Answer

    \(s^2\sqrt[4]{s^2}\)

    Example \(\PageIndex{112}\)

    \(\sqrt[3]{108x^5}\)

    Example \(\PageIndex{113}\)

    \(\sqrt[4]{48y^6}\)

    Answer

    \(2y\sqrt[4]{3y^2}\)

    Example \(\PageIndex{114}\)

    \(\sqrt[5]{−486}\)

    Example \(\PageIndex{115}\)

    \(\sqrt[6]{−64}\)

    Answer

    not real

    Example \(\PageIndex{116}\)

    \(\frac{\sqrt[4]{64}}{\sqrt[4]{2}}\)

    Example \(\PageIndex{117}\)

    \(\frac{\sqrt[5]{128x^8}}{\sqrt[5]{2x^2}}\)

    Answer

    \(2x\sqrt[5]{2x}\)

    Example \(\PageIndex{118}\)

    \(\sqrt[5]{\frac{96r^{11}}{s^3}}\)

    Example \(\PageIndex{119}\)

    \(\sqrt[6]{\frac{128u^7}{v^3}}\)

    Answer

    \(\frac{2u^3\sqrt[6]{2uv^3}}{v}\)

    Example \(\PageIndex{120}\)

    \(\sqrt[3]{81}−\sqrt[3]{192}\)

    Example \(\PageIndex{121}\)

    \(\sqrt[4]{512}−\sqrt[4]{32}\)

    Answer

    \(4\sqrt[4]{2}\)

    Example \(\PageIndex{122}\)

    \(\sqrt[3]{64a^{10}}−\sqrt[3]{−216a^{12}}\)

    Example \(\PageIndex{123}\)

    \(\sqrt[4]{486u^7}+\sqrt[4]{768u^3}\)

    Answer

    \(3u\sqrt[4]{6u^3}+4\sqrt[4]{3u^3}\)

    Everyday Math

    Example \(\PageIndex{124}\)

    Population growth The expression \(10·x^n\) models the growth of a mold population after n generations. There were 10 spores at the start, and each had x offspring. So \(10·x^n\) is the number of offspring at the fifth generation. At the fifth generation there were 10,240 offspring. Simplify the expression \(\sqrt[5]{\frac{10,240}{10}}\) to determine the number of offspring of each spore.

    Example \(\PageIndex{125}\)

    Spread of a virus The expression \(3·x^n\) models the spread of a virus after n cycles. There were three people originally infected with the virus, and each of them infected x people. So \(3·x^4\) is the number of people infected on the fourth cycle. At the fourth cycle 1875 people were infected. Simplify the expression \(\sqrt[4]{\frac{1875}{3}}\) to determine the number of people each person infected.

    Answer

    5

    Writing Exercises

    Example \(\PageIndex{126}\)

    Explain how you know that \(\sqrt[5]{x^{10}}=x^2\).

    Example \(\PageIndex{127}\)

    Explain why \(\sqrt[4]{−64}\) is not a real number but \(\sqrt[3]{−64}\) is.

    Answer

    Answers may vary.

    Self Check

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

    This table has four columns and five rows. The first row labels each column: “I can…,” “Confidentaly,” “With some help,” and “No – I don’t get it!” The rows under the “I can…,” column read, “simplify expressions with hither roots.,” “use the product property to simplify expressions with higher roots.,” “use the quotient property to simplify expressions with higher roots.,” and “add and subtract higher roots.” The rest of the rows under the columns are empty.

    ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

    This page titled 8.7E: Exercises is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

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