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8.1 Simplify Expressions with Roots

Square roots and higher-index roots, why even indices force absolute-value bars on variables, and estimating irrational roots between consecutive integers.

Intermediate Algebra 2e, 8.1 Simplify Expressions with Roots

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65 exercises

8.1 Exercise 1

Before you get started, take this readiness quiz.

Simplify: ⓐ (−9)2\left(−9\right)^{2} ⓑ −92− 9^{2} ⓒ (−9)3.\left(−9\right)^{3} .
If you missed this problem, review the reference.

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8.1 Exercise 2

Round 3.8463.846 to the nearest hundredth.
If you missed this problem, review the reference.

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8.1 Exercise 3

Simplify: ⓐ x3·x3x^{3} \cdot x^{3} ⓑ y2·y2·y2y^{2} \cdot y^{2} \cdot y^{2} ⓒ z3·z3·z3·z3.z^{3} \cdot z^{3} \cdot z^{3} \cdot z^{3} .
If you missed this problem, review the reference.

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8.1 Exercise 4

Simplify: ⓐ 144\sqrt{144} ⓑ −289.− \sqrt{289} .

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8.1 Exercise 5

Simplify: ⓐ −64− \sqrt{64} ⓑ 225.\sqrt{225} .

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8.1 Exercise 6

Simplify: ⓐ 100\sqrt{100} ⓑ −121.− \sqrt{121} .

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8.1 Exercise 7

Simplify: ⓐ −196\sqrt{−196} ⓑ −64.− \sqrt{64} .

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8.1 Exercise 8

Simplify: ⓐ −169\sqrt{−169} ⓑ −81.− \sqrt{81} .

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8.1 Exercise 9

Simplify: ⓐ −49− \sqrt{49} ⓑ −121.\sqrt{−121} .

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8.1 Exercise 10

Simplify: ⓐ 643\sqrt[3]{64} ⓑ 814\sqrt[4]{81} ⓒ 325.\sqrt[5]{32} .

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8.1 Exercise 11

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

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8.1 Exercise 12

Simplify: ⓐ 10003\sqrt[3]{1000} ⓑ 164\sqrt[4]{16} ⓒ 10245.\sqrt[5]{1024} .

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8.1 Exercise 13

Simplify: ⓐ −1253\sqrt[3]{−125} ⓑ -164\sqrt[4]{- 16} ⓒ −2435.\sqrt[5]{−243} .

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8.1 Exercise 14

Simplify: ⓐ −273\sqrt[3]{−27} ⓑ −2564\sqrt[4]{−256} ⓒ −325.\sqrt[5]{−32} .

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8.1 Exercise 15

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

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8.1 Exercise 17

Estimate each root between two consecutive whole numbers:

ⓐ 38\sqrt{38} ⓑ 933\sqrt[3]{93}

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8.1 Exercise 18

Estimate each root between two consecutive whole numbers:

ⓐ 84\sqrt{84} ⓑ 1523\sqrt[3]{152}

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8.1 Exercise 19

Round to two decimal places: ⓐ 17\sqrt{17} ⓑ 493\sqrt[3]{49} ⓒ 514.\sqrt[4]{51} .

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8.1 Exercise 20

Round to two decimal places:

ⓐ 11\sqrt{11} ⓑ 713\sqrt[3]{71} ⓒ 1274.\sqrt[4]{127} .

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8.1 Exercise 21

Round to two decimal places:

ⓐ 13\sqrt{13} ⓑ 843\sqrt[3]{84} ⓒ 984.\sqrt[4]{98} .

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8.1 Exercise 22

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

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8.1 Exercise 23

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

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8.1 Exercise 24

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

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8.1 Exercise 25

Simplify: ⓐ x6\sqrt{x^{6}} ⓑ y16.\sqrt{y^{16}} .

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8.1 Exercise 26

Simplify: ⓐ y18\sqrt{y^{18}} ⓑ z12.\sqrt{z^{12}} .

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8.1 Exercise 27

Simplify: ⓐ m4\sqrt{m^{4}} ⓑ b10.\sqrt{b^{10}} .

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8.1 Exercise 28

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

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8.1 Exercise 29

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

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8.1 Exercise 30

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

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8.1 Exercise 31

Simplify: ⓐ 16n2\sqrt{16 n^{2}} ⓑ −81c2.− \sqrt{81 c^{2}} .

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8.1 Exercise 32

Simplify: ⓐ 64x2\sqrt{64 x^{2}} ⓑ −100p2.− \sqrt{100 p^{2}} .

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8.1 Exercise 33

Simplify: ⓐ 169y2\sqrt{169 y^{2}} ⓑ −121y2.− \sqrt{121 y^{2}} .

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8.1 Exercise 34

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

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8.1 Exercise 35

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

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8.1 Exercise 36

Simplify: ⓐ 125q93\sqrt[3]{125 q^{9}} ⓑ 243q255.\sqrt[5]{243 q^{25}} .

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8.1 Exercise 37

Simplify: ⓐ 36x2y2\sqrt{36 x^{2} y^{2}} ⓑ 121a6b8\sqrt{121 a^{6} b^{8}} ⓒ 64p63q93.\sqrt[3]{64 p^{63} q^{9}} .

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8.1 Exercise 38

Simplify: ⓐ 100a2b2\sqrt{100 a^{2} b^{2}} ⓑ 144p12q20\sqrt{144 p^{12} q^{20}} ⓒ 8x30y123\sqrt[3]{8 x^{30} y^{12}}

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8.1 Exercise 39

Simplify: ⓐ 225m2n2\sqrt{225 m^{2} n^{2}} ⓑ 169x10y14\sqrt{169 x^{10} y^{14}} ⓒ 27w36z153\sqrt[3]{27 w^{36} z^{15}}

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8.1 Exercise 40

Simplify.

ⓐ 64\sqrt{64} ⓑ −81− \sqrt{81}

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8.1 Exercise 42

Simplify.

ⓐ 196\sqrt{196} ⓑ −1− \sqrt{1}

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8.1 Exercise 44

Simplify.

ⓐ 49\sqrt{\frac{4}{9}} ⓑ −0.01− \sqrt{0.01}

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8.1 Exercise 46

Simplify.

ⓐ −121\sqrt{−121} ⓑ −289− \sqrt{289}

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8.1 Exercise 48

Simplify.

ⓐ −225− \sqrt{225} ⓑ −9\sqrt{−9}

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8.1 Exercise 50

Simplify.

ⓐ 2163\sqrt[3]{216} ⓑ 2564\sqrt[4]{256}

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8.1 Exercise 52

Simplify.

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

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8.1 Exercise 54

Simplify.

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

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8.1 Exercise 56

Simplify.

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

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8.1 Exercise 58

Estimate each root between two consecutive whole numbers.

ⓐ 70\sqrt{70} ⓑ 713\sqrt[3]{71}

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8.1 Exercise 60

Estimate each root between two consecutive whole numbers.

ⓐ 200\sqrt{200} ⓑ 1373\sqrt[3]{137}

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8.1 Exercise 62

Approximate each root and round to two decimal places.

ⓐ 19\sqrt{19} ⓑ 893\sqrt[3]{89} ⓒ 974\sqrt[4]{97}

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8.1 Exercise 64

Approximate each root and round to two decimal places.

ⓐ 53\sqrt{53} ⓑ 1473\sqrt[3]{147} ⓒ 4524\sqrt[4]{452}

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8.1 Exercise 66

Simplify using absolute values as necessary.

ⓐ u55\sqrt[5]{u^{5}} ⓑ v88\sqrt[8]{v^{8}}

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8.1 Exercise 68

Simplify using absolute values as necessary.

ⓐ y44\sqrt[4]{y^{4}} ⓑ m77\sqrt[7]{m^{7}}

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8.1 Exercise 70

Simplify using absolute values as necessary.

ⓐ x6\sqrt{x^{6}} ⓑ y16\sqrt{y^{16}}

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8.1 Exercise 72

Simplify using absolute values as necessary.

ⓐ x24\sqrt{x^{24}} ⓑ y22\sqrt{y^{22}}

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8.1 Exercise 74

Simplify using absolute values as necessary.

ⓐ x93\sqrt[3]{x^{9}} ⓑ y124\sqrt[4]{y^{12}}

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8.1 Exercise 76

Simplify using absolute values as necessary.

ⓐ m84\sqrt[4]{m^{8}} ⓑ n205\sqrt[5]{n^{20}}

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8.1 Exercise 78

Simplify using absolute values as necessary.

ⓐ 49x2\sqrt{49 x^{2}} ⓑ −81x18− \sqrt{81 x^{18}}

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8.1 Exercise 80

Simplify using absolute values as necessary.

ⓐ 121m20\sqrt{121 m^{20}} ⓑ −64a2− \sqrt{64 a^{2}}

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8.1 Exercise 82

Simplify using absolute values as necessary.

ⓐ 16x84\sqrt[4]{16 x^{8}} ⓑ 64y126\sqrt[6]{64 y^{12}}

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8.1 Exercise 84

Simplify using absolute values as necessary.

ⓐ 216a63\sqrt[3]{216 a^{6}} ⓑ 32b205\sqrt[5]{32 b^{20}}

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8.1 Exercise 86

Simplify using absolute values as necessary.

ⓐ 144x2y2\sqrt{144 x^{2} y^{2}} ⓑ 169w8y10\sqrt{169 w^{8} y^{10}} ⓒ 8a51b63\sqrt[3]{8 a^{51} b^{6}}

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8.1 Exercise 88

Simplify using absolute values as necessary.

ⓐ 121a2b2\sqrt{121 a^{2} b^{2}} ⓑ 9c8d12\sqrt{9 c^{8} d^{12}} ⓒ 64x15y663\sqrt[3]{64 x^{15} y^{66}}

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8.1 Exercise 90

Why is there no real number equal to −64?\sqrt{−64} ?

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8.1 Exercise 92

Explain what is meant by the nth root of a number.

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