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9.7 Higher Roots

Cube roots and beyond, where an odd index tolerates a negative radicand and an even one does not, with the same product, quotient, and like-term moves.

Elementary Algebra 2e, 9.7 Higher Roots

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86 exercises

9.7 Exercise 1

Before you get started, take this readiness quiz.

Simplify: y5y4y^{5} y^{4}.
If you missed this problem, review the reference.

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9.7 Exercise 2

Simplify: (n2)6\left(n^{2}\right)^{6}.
If you missed this problem, review the reference.

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9.7 Exercise 3

Simplify: x8x3\frac{x^{8}}{x^{3}}.
If you missed this problem, review the reference.

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9.7 Exercise 4

Simplify: ⓐ 83\sqrt[3]{8} ⓑ 814\sqrt[4]{81} ⓒ 325\sqrt[5]{32}.

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9.7 Exercise 5

Simplify: ⓐ 273\sqrt[3]{27} ⓑ 2564\sqrt[4]{256} ⓒ 2435\sqrt[5]{243}.

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9.7 Exercise 6

Simplify: ⓐ 10003\sqrt[3]{1000} ⓑ 164\sqrt[4]{16} ⓒ 325\sqrt[5]{32}.

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9.7 Exercise 7

Simplify: ⓐ −643\sqrt[3]{−64} ⓑ −164\sqrt[4]{−16} ⓒ −2435\sqrt[5]{−243}.

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9.7 Exercise 8

Simplify: ⓐ −1253\sqrt[3]{−125} ⓑ −164\sqrt[4]{−16} ⓒ −325\sqrt[5]{−32}.

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9.7 Exercise 9

Simplify: ⓐ −2163\sqrt[3]{−216} ⓑ −814\sqrt[4]{−81} ⓒ −10245\sqrt[5]{−1024}.

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9.7 Exercise 10

Simplify: ⓐ x2\sqrt{x^{2}} ⓑ n33\sqrt[3]{n^{3}} ⓒ p44\sqrt[4]{p^{4}} ⓓ y55\sqrt[5]{y^{5}}.

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9.7 Exercise 11

Simplify: ⓐ b2\sqrt{b^{2}} ⓑ w33\sqrt[3]{w^{3}} ⓒ m44\sqrt[4]{m^{4}} ⓓ q55\sqrt[5]{q^{5}}.

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9.7 Exercise 12

Simplify: ⓐ y2\sqrt{y^{2}} ⓑ p33\sqrt[3]{p^{3}} ⓒ z44\sqrt[4]{z^{4}} ⓓ q55\sqrt[5]{q^{5}}.

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9.7 Exercise 13

Simplify: ⓐ y183\sqrt[3]{y^{18}} ⓑ z84\sqrt[4]{z^{8}}.

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9.7 Exercise 14

Simplify: ⓐ u124\sqrt[4]{u^{12}} ⓑ v153\sqrt[3]{v^{15}}.

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9.7 Exercise 15

Simplify: ⓐ c205\sqrt[5]{c^{20}} ⓑ d246\sqrt[6]{d^{24}}.

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9.7 Exercise 16

Simplify: ⓐ 64p63\sqrt[3]{64 p^{6}} ⓑ 16q124\sqrt[4]{16 q^{12}}.

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9.7 Exercise 17

Simplify: ⓐ 27x273\sqrt[3]{27 x^{27}} ⓑ 81q284\sqrt[4]{81 q^{28}}.

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9.7 Exercise 18

Simplify: ⓐ 125p93\sqrt[3]{125 p^{9}} ⓑ 243q255\sqrt[5]{243 q^{25}}.

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9.7 Exercise 19

Simplify: ⓐ x43\sqrt[3]{x^{4}} ⓑ x74\sqrt[4]{x^{7}}.

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9.7 Exercise 20

Simplify: ⓐ y64\sqrt[4]{y^{6}} ⓑ z53\sqrt[3]{z^{5}}.

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9.7 Exercise 21

Simplify: ⓐ p85\sqrt[5]{p^{8}} ⓑ q136\sqrt[6]{q^{13}}.

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9.7 Exercise 22

Simplify: ⓐ 163\sqrt[3]{16} ⓑ 2434\sqrt[4]{243}.

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9.7 Exercise 23

Simplify: ⓐ 813\sqrt[3]{81} ⓑ 644\sqrt[4]{64}.

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9.7 Exercise 24

Simplify: ⓐ 6253\sqrt[3]{625} ⓑ 7294\sqrt[4]{729}.

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9.7 Exercise 25

Simplify: ⓐ 24x73\sqrt[3]{24 x^{7}} ⓑ 80y144\sqrt[4]{80 y^{14}}.

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9.7 Exercise 26

Simplify: ⓐ 54p103\sqrt[3]{54 p^{10}} ⓑ 64q104\sqrt[4]{64 q^{10}}.

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9.7 Exercise 27

Simplify: ⓐ 128m113\sqrt[3]{128 m^{11}} ⓑ 162n74\sqrt[4]{162 n^{7}}.

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9.7 Exercise 28

Simplify: ⓐ −273\sqrt[3]{−27} ⓑ −164\sqrt[4]{−16}.

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9.7 Exercise 29

Simplify: ⓐ −1083\sqrt[3]{−108} ⓑ −484\sqrt[4]{−48}.

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9.7 Exercise 30

Simplify: ⓐ −6253\sqrt[3]{−625} ⓑ −3244\sqrt[4]{−324}.

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9.7 Exercise 31

Simplify: ⓐ a8a53\sqrt[3]{\frac{a^{8}}{a^{5}}} ⓑ a10a24\sqrt[4]{\frac{a^{10}}{a^{2}}}.

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9.7 Exercise 32

Simplify: ⓐ x7x34\sqrt[4]{\frac{x^{7}}{x^{3}}} ⓑ y17y54\sqrt[4]{\frac{y^{17}}{y^{5}}}.

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9.7 Exercise 33

Simplify: ⓐ m13m73\sqrt[3]{\frac{m^{13}}{m^{7}}} ⓑ n12n25\sqrt[5]{\frac{n^{12}}{n^{2}}}.

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9.7 Exercise 34

Simplify: ⓐ −108323\frac{\sqrt[3]{−108}}{\sqrt[3]{2}} ⓑ 96x743x24\frac{\sqrt[4]{96 x^{7}}}{\sqrt[4]{3 x^{2}}}.

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9.7 Exercise 35

Simplify: ⓐ −532323\frac{\sqrt[3]{−532}}{\sqrt[3]{2}} ⓑ 486m1143m54\frac{\sqrt[4]{486 m^{11}}}{\sqrt[4]{3 m^{5}}}.

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9.7 Exercise 36

Simplify: ⓐ −192333\frac{\sqrt[3]{−192}}{\sqrt[3]{3}} ⓑ 324n742n34\frac{\sqrt[4]{324 n^{7}}}{\sqrt[4]{2 n^{3}}}.

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9.7 Exercise 37

Simplify: ⓐ 24x7y33\sqrt[3]{\frac{24 x^{7}}{y^{3}}} ⓑ 48x10y84\sqrt[4]{\frac{48 x^{10}}{y^{8}}}.

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9.7 Exercise 38

Simplify: ⓐ 108c10d63\sqrt[3]{\frac{108 c^{10}}{d^{6}}} ⓑ 80x10y54\sqrt[4]{\frac{80 x^{10}}{y^{5}}}.

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9.7 Exercise 39

Simplify: ⓐ 40r3s3\sqrt[3]{\frac{40 r^{3}}{s}} ⓑ 162m14n124\sqrt[4]{\frac{162 m^{14}}{n^{12}}}.

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9.7 Exercise 40

Simplify: ⓐ 4x3+4x3\sqrt[3]{4 x} + \sqrt[3]{4 x} ⓑ 484−2844 \sqrt[4]{8} - 2 \sqrt[4]{8}.

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9.7 Exercise 41

Simplify: ⓐ 3x5+3x5\sqrt[5]{3 x} + \sqrt[5]{3 x} ⓑ 393−933 \sqrt[3]{9} - \sqrt[3]{9}.

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9.7 Exercise 42

Simplify: ⓐ 10y4+10y4\sqrt[4]{10 y} + \sqrt[4]{10 y} ⓑ 5326−33265 \sqrt[6]{32} - 3 \sqrt[6]{32}.

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9.7 Exercise 43

Simplify: ⓐ 543−163\sqrt[3]{54} - \sqrt[3]{16} ⓑ 484+2434\sqrt[4]{48} + \sqrt[4]{243}.

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9.7 Exercise 44

Simplify: ⓐ 1923−813\sqrt[3]{192} - \sqrt[3]{81} ⓑ 324+5124\sqrt[4]{32} + \sqrt[4]{512}.

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9.7 Exercise 45

Simplify: ⓐ 1283−2503\sqrt[3]{128} - \sqrt[3]{250} ⓑ 645+4865\sqrt[5]{64} + \sqrt[5]{486}.

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9.7 Exercise 46

Simplify: ⓐ 24x43−−81x73\sqrt[3]{24 x^{4}} - \sqrt[3]{−81 x^{7}} ⓑ 162y94+512y54\sqrt[4]{162 y^{9}} + \sqrt[4]{512 y^{5}}.

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9.7 Exercise 47

Simplify: ⓐ 32y53−−108y83\sqrt[3]{32 y^{5}} - \sqrt[3]{−108 y^{8}} ⓑ 243r114+768r104\sqrt[4]{243 r^{11}} + \sqrt[4]{768 r^{10}}.

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9.7 Exercise 48

Simplify: ⓐ 40z73−−135z43\sqrt[3]{40 z^{7}} - \sqrt[3]{−135 z^{4}} ⓑ 80s134+1280s64\sqrt[4]{80 s^{13}} + \sqrt[4]{1280 s^{6}}.

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9.7 Exercise 50

Simplify.

ⓐ 273\sqrt[3]{27} ⓑ 164\sqrt[4]{16} ⓒ 2435\sqrt[5]{243}

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9.7 Exercise 52

Simplify.

ⓐ 1253\sqrt[3]{125} ⓑ 12964\sqrt[4]{1296} ⓒ 10245\sqrt[5]{1024}

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9.7 Exercise 54

Simplify.

ⓐ −643\sqrt[3]{−64} ⓑ −164\sqrt[4]{−16} ⓒ −2435\sqrt[5]{−243}

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9.7 Exercise 56

Simplify.

ⓐ −5123\sqrt[3]{−512} ⓑ −814\sqrt[4]{−81} ⓒ −15\sqrt[5]{−1}

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9.7 Exercise 60

Simplify.

ⓐ k88\sqrt[8]{k^{8}} ⓑ p66\sqrt[6]{p^{6}}

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9.7 Exercise 62

Simplify.

ⓐ a105\sqrt[5]{a^{10}} ⓑ b273\sqrt[3]{b^{27}}

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9.7 Exercise 64

Simplify.

ⓐ r126\sqrt[6]{r^{12}} ⓑ s303\sqrt[3]{s^{30}}

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9.7 Exercise 66

Simplify.

ⓐ −8c93\sqrt[3]{−8 c^{9}} ⓑ 125d153\sqrt[3]{125 d^{15}}

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9.7 Exercise 68

Simplify.

ⓐ 128r147\sqrt[7]{128 r^{14}} ⓑ 81s244\sqrt[4]{81 s^{24}}

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9.7 Exercise 70

Simplify.

ⓐ u75\sqrt[5]{u^{7}} ⓑ v116\sqrt[6]{v^{11}}

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9.7 Exercise 72

Simplify.

ⓐ p85\sqrt[5]{p^{8}} ⓑ q83\sqrt[3]{q^{8}}

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9.7 Exercise 74

Simplify.

ⓐ 6253\sqrt[3]{625} ⓑ 1286\sqrt[6]{128}

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9.7 Exercise 76

Simplify.

ⓐ 31254\sqrt[4]{3125} ⓑ 813\sqrt[3]{81}

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9.7 Exercise 78

Simplify.

ⓐ 96a75\sqrt[5]{96 a^{7}} ⓑ 375b43\sqrt[3]{375 b^{4}}

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9.7 Exercise 80

Simplify.

ⓐ 512p53\sqrt[3]{512 p^{5}} ⓑ 324q74\sqrt[4]{324 q^{7}}

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9.7 Exercise 82

Simplify.

ⓐ −4865\sqrt[5]{−486} ⓑ −646\sqrt[6]{−64}

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9.7 Exercise 84

Simplify.

ⓐ −83\sqrt[3]{−8} ⓑ −164\sqrt[4]{−16}

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9.7 Exercise 86

Simplify.

ⓐ d12d75\sqrt[5]{\frac{d^{12}}{d^{7}}} ⓑ m12m48\sqrt[8]{\frac{m^{12}}{m^{4}}}

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9.7 Exercise 88

Simplify.

ⓐ r14r53\sqrt[3]{\frac{r^{14}}{r^{5}}} ⓑ c21c94\sqrt[4]{\frac{c^{21}}{c^{9}}}

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9.7 Exercise 90

Simplify.

ⓐ −625353\frac{\sqrt[3]{−625}}{\sqrt[3]{5}} ⓑ 80m745m4\frac{\sqrt[4]{80 m^{7}}}{\sqrt[4]{5 m}}

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9.7 Exercise 92

Simplify.

ⓐ 16263\sqrt[3]{\frac{162}{6}} ⓑ 160r105r34\sqrt[4]{\frac{160 r^{10}}{5 r^{3}}}

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9.7 Exercise 94

Simplify.

ⓐ 96r11s35\sqrt[5]{\frac{96 r^{11}}{s^{3}}} ⓑ 128u7v36\sqrt[6]{\frac{128 u^{7}}{v^{3}}}

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9.7 Exercise 96

Simplify.

ⓐ 625u10v33\sqrt[3]{\frac{625 u^{10}}{v^{3}}} ⓑ 729c21d84\sqrt[4]{\frac{729 c^{21}}{d^{8}}}

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9.7 Exercise 98

Simplify.

ⓐ 15q3+15q3\sqrt[3]{15 q} + \sqrt[3]{15 q} ⓑ 2274−62742 \sqrt[4]{27} - 6 \sqrt[4]{27}

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9.7 Exercise 102

Simplify.

ⓐ 2503−543\sqrt[3]{250} - \sqrt[3]{54} ⓑ 2434−18754\sqrt[4]{243} - \sqrt[4]{1875}

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9.7 Exercise 104

Simplify.

ⓐ 2434+12504\sqrt[4]{243} + \sqrt[4]{1250} ⓑ 20003+543\sqrt[3]{2000} + \sqrt[3]{54}

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9.7 Exercise 106

Simplify.

ⓐ 80b53−−270b33\sqrt[3]{80 b^{5}} - \sqrt[3]{−270 b^{3}} ⓑ 160v104−1280v34\sqrt[4]{160 v^{10}} - \sqrt[4]{1280 v^{3}}

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9.7 Exercise 108

Simplify.

646\sqrt[6]{64}

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9.7 Exercise 112

Simplify.

125d153\sqrt[3]{125 d^{15}}

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9.7 Exercise 114

Simplify.

s104\sqrt[4]{s^{10}}

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9.7 Exercise 116

Simplify.

48y64\sqrt[4]{48 y^{6}}

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9.7 Exercise 118

Simplify.

−646\sqrt[6]{−64}

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9.7 Exercise 120

Simplify.

128x852x25\frac{\sqrt[5]{128 x^{8}}}{\sqrt[5]{2 x^{2}}}

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9.7 Exercise 122

Simplify.

128u7v36\sqrt[6]{\frac{128 u^{7}}{v^{3}}}

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9.7 Exercise 124

Simplify.

5124−324\sqrt[4]{512} - \sqrt[4]{32}

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9.7 Exercise 126

Simplify.

486u74+768u34\sqrt[4]{486 u^{7}} + \sqrt[4]{768 u^{3}}

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9.7 Exercise 128

Spread of a virus The expression 3·xn3 \cdot x^{n} models the spread of a virus after nn cycles. There were three people originally infected with the virus, and each of them infected xx people. So 3·x43 \cdot x^{4} is the number of people infected on the fourth cycle. At the fourth cycle 1875 people were infected. Simplify the expression 187534\sqrt[4]{\frac{1875}{3}} to determine the number of people each person infected.

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9.7 Exercise 130

Explain why −644\sqrt[4]{−64} is not a real number but −643\sqrt[3]{−64} is.

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Adapted from Elementary Algebra 2e © 2020 Rice University. Licensed under CC BY 4.0. Source: Elementary Algebra 2e. Imported at revision 77a933c.