chapter 8 roots and radicals - way cool …€¦ · · 2014-09-11chapter eight additional...

Chapter Eight Additional Exercises 209

CHAPTER 8 ROOTS AND RADICALS Section 8.1 Evaluating Roots Objective 1 Find square roots. Find all square roots of the number. 1. 81 2. 169 3. 256 4. 400 5. 324 6. 576 7. 361 8. 1225

9. 64

49 10.

144

121 11.

625

169 12.

289

196

Find the square root.

13. 100 14. 144− 15. 1− 16. 36

17. 225 18. 900− 19. 000,10 20. 529−

21. 225

64 22.

441

169− 23. 361

256 24.

6400

2500−

Objective 2 Decide whether a given root is rational, irrational, or not a real number. Tell whether the square root is rational, irrational, or not a real number.

25. 144 26. 3 27. 16− 28. 25−

29. 95− 30. 43− 31. 1600 32. 200

33. 49

4 34.

484

625− 35. 5.2 36. 0.9−

210 Chapter Eight Additional Exercises

Objective 3 Find decimal approximations for irrational square roots. Use a calculator to find a decimal approximation for the root. Round answers to the nearest thousandth.

37. 45 38. 31 39. 15− 40. 90−

41. 150− 42. 300 43. 131− 44. 245

45. 832 46. 1100− 47. 500,12 48. 358,26− Objective 4 Use the Pythagorean formula. Find the length of the unknown side of the right triangle with sides a, b, and c, where c is the hypotenuse. If necessary, round your answer to the nearest thousandth. 49. 15,8 == ba 50. 21,28 == ba 51. 25,35 == ac 52. 24,25 == bc 53. 16,17 == ac 54. 8,3 == ba Use the Pythagorean formula to solve the problem. If necessary, round your answer to the nearest thousandth. 55. The hypotenuse of a right triangle measure 10 centimeters and one leg measures 4

centimeters. How long is the other leg? 56. Two sides of a rectangle measure 7 centimeters and 11 centimeters. How long are the

diagonals of the rectangle? 57. A diagonal of a rectangle measures 29 inches. The length of the rectangle is 21 inches.

Find the width of the rectangle. 58. A diagonal of a rectangle measures 2.5 meters. The width of the rectangle is 1.5 meters.

Find the length of the rectangle.


59. A ladder 25 feet long leans against a wall. The foot of the ladder is 7 feet from the base of the wall. How high on the wall does the top of the ladder rest?

60. Laura is flying a kite on 60 feet of string. How high is the kite above her hand (vertically) if the horizontal distance between Laura and the kite is 36 feet?

61. Kevin started to drive due south at the same time Lydia started to drive due west. Lydia

drove 21 miles in the same time that Kevin drove 28 miles. How far apart were they at that time?

62. A cable from the top of a pole 15 feet tall is pulled taut and attached to the ground 8 feet

from the base of the pole. How long is the cable?


63. The foot of a loading ramp 9 feet long is placed 7 feet from the base of a platform. The top of the ramp rests on the platform. How high is the platform?

64. A plane flies due east for 35 miles and then due south until it is 37 miles from its starting

point. How far south did the plane fly?

Objective 5 Use the distance formula. Find the distance between each pair of points. 65. ( ) ( )8,3,6,6 66. ( ) ( )1,9,2,6 67. ( ) ( )2,0,4,3 − 68. ( ) ( )6,5,6,5 −− 69. ( ) ( )7,4,1,5 −−− 70. ( ) ( )3,4,0,0 − 71. ( ) ( )6,7,6,7 −−− 72. ( ) ( )4,0,0,4 73. ( ) ( )1,6,2,2 −− 74. ( ) ( )1,2,8,5 −− 75. ( ) ( )3,2,2,3 − 76. ( ) ( )2,9,11,7 − Objective 6 Find cube, fourth, and other roots. Find each root that is a real number.

77. 3 27 78. 3 125− 79. 5 1− 80. 3 1000−

81. 3 216− 82. 3 8−− 83. 5 243− 84. 3 64−−

85. 4 81− 86. 3 343− 87. 4 625− 88. 5 32−−

89. 6 1−− 90. 5 000,100− 91. 4 1−− 92. 3 512


Mixed Exercises Find each root that is a real number.

93. 196 94. 7 1− 95. 144− 96. 4 256

97. 6 64− 98. 4 81−− 99. 289

225− 100. 3

125

27−−

Write rational, irrational, or not a real number to describe the number. If the number is rational, give its exact value. If the number is irrational, give a decimal approximation to the nearest thousandth. Use a calculator as necessary.

101. 225 102. 7 103. 49− 104. 75

105. 169

81− 106. 441

361 107. 5.8 108. 0.09−

Find the length of the unknown side of the right triangle with sides a, b, and c, where c is the hypotenuse. If necessary, round your answer to the nearest thousandth. 109. 24,10 == ba 110. 9,5 == cb 111. 12,1 == ca 112. 40,41 == ac 113. 8,5 == ba 114. 11,8 == cb 115. 17,9 == ca 116. 11,6 == cb Writing/Conceptual Exercises Answer the question. 117. How many fourth roots does 0 have? 118. How many real number fourth roots does any positive number have? 119. How many real number fourth roots does any negative number have?


120. Which of the following numbers have rational square roots?

(a) 5 (b) 49 (c) 9− (d) 121

121. Which of the following numbers have irrational square roots?

(a) 36

25 (b)

2

1 (c) 15 (d)

9

4−

What must be true about b for the statement to be true? 122. 3 b is a negative number. 123. 3 b− is a negative number.

124. 3 b− is a positive number. 125. 3 b− is a positive number.

126. 3 b is zero. Section 8.2 Multiplying, Dividing, and Simplifying Radicals Objective 1 Multiply square root radicals. Use the product rule for radicals to find the product.

127. 35 ⋅ 128. 312 ⋅

129. 77 ⋅ 130. 322 ⋅

131. 76 ⋅ 132. 246 ⋅

133. 2323 ⋅ 134. 205 ⋅

135. 1113 ⋅ 136. 0,6 >⋅ rr

137. 0,72 >⋅ xx 138. 0,0,218 >>⋅ baba


Objective 2 Simplify radicals by using the product rule. Simplify the radical. Assume that all variables represent nonnegative real numbers.

139. 24 140. 72 141. 500 142. 150−

143. 56− 144. 288 145. 542− 146. 10003

147. 3y 148. 42qp 149. 580a 150. 67125 sr

Find the product and simplify.

151. 155 ⋅ 152. 203 ⋅ 153. 126 ⋅

154. 1030 ⋅ 155. 2211 ⋅ 156. 2418 ⋅ Objective 3 Simplify radicals by using the quotient rule. Use the quotient rule and product rule, as necessary, to simplify the expression.

157. 121

169 158.

49

13 159.

35

753

160. 23

2009 161.

34

48 162. 42

7

6 ⋅

163. 2

1

18

11 ⋅ 164. 2

5

8

3 ⋅ 165. 27

16

3

5 ⋅

Objective 4 Simplify radicals involving variables.

166. 84x 167. 33p 168. 5

4

x

y

169. 620a 170. 2

27

a 171. 48x

172. 3 781x 173. 4

3

8

x 174. 3

6

64

x


Objective 5 Simplify higher roots. Simplify the expression.

175. 3 24 176. 3 81 177. 3 128

178. 4 243 179. 5 64 180. 3

27

64−

181. 3

216

1−− 182. 4

81

16− 183. 4

256

625

184. 33 93 ⋅ 185. 55 12525 ⋅ 186. 44 273 ⋅ Mixed Exercises Find the product and simplify if possible.

187. 1111 ⋅ 188. 1513 ⋅ 189. 753 ⋅

190. 5010 ⋅ 191. 2613 ⋅ 192. 7979 ⋅ Simplify the expression. Assume that all variables represent positive numbers.

193. 99 194. 245 195. 175−

196. 1545 ⋅ 197. 6328 ⋅ 198. xx 39 ⋅

199. 225

144− 200. 117

3328 201. 5

32

243−

202. 3

50

a

a 203.

125

27

3

5 ⋅ 204. 10

7

63

40 ⋅

205. ba 375 206. 3 40 207. 5 64−

208. 33 42 ⋅ 209. 7698 yx 210. 4 243


Writing/Conceptual Exercises Decide whether the statement is true or false.

211. 254254 ⋅=⋅ 212. 254254 +=+

213. ( ) 1010 2 −=− 214. ( ) 10103 3 −=−

215. Which of the following radicals are simplified according to the guidelines given in the

textbook?

(a) 54 (b) 19 (c) 21 (d) 72 216. Which of the following radicals are simplified according to the guidelines given in the

textbook?

(a) 3 9 (b) 3 16 (c) 3 125 (d) 3 20

Section 8.3 Adding and Subtracting Radicals Objective 1 Add and subtract radicals. Add or subtract, as indicated.

217. 5954 +− 218. 115117 −

219. 33 1010 + 220. 7475 −

221. 35 + 222. 373235 +−

223. 66462 +− 224. 545355 −+

225. 444 122126129 −+ 226. 555 44246 +− Objective 2 Simplify radical sums and differences. Simplify and add or subtract terms wherever possible. Assume that all variables represent nonnegative real numbers.

227. 238 + 228. 2322 +

229. 1862003 − 230. 445113 −


231. 203542245 +− 232. 28428275 +−−

233. 122

350

5

1 − 234. 324

5

2

8 −

235. 484

372

6

5 − 236. 82

3128

8

3 −

237. zz 18386 − 238. ww 62963 −

239. xx 500203 − 240. 44 32162 −

241. 44 4832432 − 242. 3 53 5 2734 rr + Objective 3 Simplify more complicated radical expressions. Perform the indicated operations. Assume that all variables represent nonnegative real numbers.

243. 15335 +⋅ 244. 422674 −⋅

245. 272147 ⋅− 246. 754358 ⋅+

247. xx 443 +⋅ 248. yy 651823 −⋅

249. xxx 3153510 ⋅− 250. 24830511 www −⋅

251. kkkk 236 2 +⋅ 252. yy 8023573 −⋅

Mixed Exercises Simplify and perform the indicated operations. Assume that all variables represent nonnegative real numbers.

253. 192193 − 254. 333 117115113 −+

255. 27312236 −+ 256. 45280 +

257. mm 73634 − 258. 33 544163 −


259. m73117 −⋅ 260. 22 8405 rr −⋅

261. 122

148

4

3 − 262. 5010

332

8

5 +

263. 286482186 −+ 264. 33 25021283 +

265. 52513133 +− 266. 1627325182 +−

267. 33 4031356 + 268. 44 4822433 −−

269. 2456863 −− 270. 328298 +−

271. 333 40135625 −+ 272. 33 24812 +−

273. zzz 322383 ++ 274. 333 1922481 ++

275. yyy 271248 ++ 276. 3 3 32 16 54 16r r r+ −

277. 275483 + 278. xxx 259100 +− Writing/Conceptual Exercises 279. Write an equation showing how the distributive property is used to justify the statement

7127874 =+ . 280. Write an equation showing how the distributive property is used to justify the statement

333 3734311 =− . Despite the fact that 25 and 3 8 are radicals that have different root indexes, they can be added to obtain a single term: 725825 3 =+=+ . 281. Make up a similar sum of radicals that leads to an answer of 12. 282. Make up a similar difference of radicals that leads to an answer of 8.


Section 8.4 Rationalizing the Denominator Objective 1 Rationalize denominators with square roots. Rationalize the denominator. Write all answers in simplest form.

283. 5

2 284.

13

4− 285.

6

2− 286.

15

5

287. 2

2 288.

10

25 289.

11

22− 290.

12

8−

291. 7

3 292.

8

5 293.

12

6 294.

28

4

295. 75

3 296.

125

5 297.

250

525 298.

96

36

Objective 2 Write radicals in simplified form. Perform the indicated operations and write all answers in simplest form. Rationalize all denominators. Assume that all variables represent positive real numbers.

299. 5

1 300.

7

6 301.

8

3

302. 27

32 303.

50

63 304.

6

5

2

3 ⋅

305. 721

8 ⋅ 306. 30

116 ⋅ 307.

35

6

5

1 ⋅

308. x

y6 309.

s

r7 310.

5

3 2ba

311. 5

2

k

mk 312.

4

3

d

df 313.

a

ba

6

20 3


Objective 3 Rationalize denominators with cube roots. Rationalize the denominator. Assume that all variables in the denominator represent nonzero real numbers.

314. 3

3

9

6 315.

3

3

4

9 316. 3

16

1

317. 3

2

5 318. 3

5

7 319. 3

3

5

320. 3

25

8 321.

3 36

6 322. 3

24

1

323. 3

80

3 324. 3

162

1 325. 3

25

4

r

326. 3

9

2

r

s 327. 3

236w

v 328. 3

25

8

u

t

Mixed Exercises Simplify the expression. Assume that all variables represent positive real numbers.

329. 7

63 330.

6

24− 331.

2

32

332. 3

5 333.

20

11 334.

14

76

335. 3

7

2 336.

22

15

5

2 ⋅ 337. 3

6

5

x

338. 3

1

98

27 ⋅ 339. 6

5

3

72

t

qt 340. 3

2

2

81

7

y

x

341. 27

14− 342. 3

98

r 343.

48

27


344. 3

15

x

t 345. 3

36

7 346.

3

23

t

y

347. 5

4162

t

x 348.

75

7 349.

28

5

x

350. 32

19 351. 3

2

3

d

c 352.

t

23

Writing/Conceptual Exercises 353. Which one of the following would be an appropriate choice for multiplying the

numerator and denominator of 3

3

7

5 in order to rationalize the denominator?

(a) 3 5 (b) 3 7 (c) 3 49 (d) 3 2

354. Which one of the following would be an appropriate choice for multiplying the

numerator and denominator of 3

3

10

4

z

y in order to rationalize the denominator?

(a) 3 10z (b) 3 100z (c) 3 210z (d) 3 2100z

355. Which one of the following would be an appropriate choice for multiplying the

numerator and denominator of 3 2

3

9

5

r

s in order to rationalize the denominator?

(a) 3 3r (b) 3 29r (c) 3 23r (d) 3 225s

356. What would be your first step in simplifying the radical 3

11

7? What property would

you be using in this step?


Section 8.5 More Simplifying and Operations with Radicals Objective 1 Simplify products of radical expressions. Simplify the expression.

357. ( )357 − 358. ( )6510 +

359. ( )32155 + 360. ( )721137 −

361. ( )( )72352 ++ 362. ( )( )23115 +−

363. ( )( )106256 +− 364. ( )( )3423324 +−

365. ( )( )105105 −+ 366. ( )( )732732 +−

367. ( )2532 − 368. ( )2

1110 −

369. ( )2543 + 370. ( )2

3223 + Objective 2 Use conjugates to rationalize denominators of radical expressions. Rationalize the denominator.

371. 23

2

− 372.

52

5

+ 373.

34

5

−

374. 3

34 + 375.

13

5

− 376.

35

2

+

377. 75

9

+ 378.

103

6

+ 379.

23

27

−+

380. 42

16

−+

381. 27

32

−+

382. 310

310

−+


Objective 3 Write radical expressions with quotients in lowest terms. Write the quotient in lowest terms.

383. 2

328 − 384.

12

726 + 385.

5

1535 +

386. 14

824 + 387.

12

269 + 388.

10

674 −

389. 2

84 + 390.

6

273 + 391.

16

1268 −

392. 28

3714 + 393.

15

453135 + 394.

24

216572 −

Mixed Exercises Simplify the expression.

395. ( )1156 − 396. ( )( )15264 +−

397. ( )2211 + 398. ( )( )137137 −+

399. 114

1

− 400.

23

25

−+

401. 17

15

+−

402. 153

55

−−

403. 2

37 − 404.

37

37

−+

405. ( )( )515352155 −+ 406. ( )22552 +

407. ( )2122 − 408. ( )( )116310 −+

409. ( )( )2525 −+ 410. 211

26

+


Write the quotient in lowest terms.

411. 8

724 − 412.

4

101020 −

413. 3

9106 − 414.

10

15115 −

415. 13

52265 − 416.

95

3133619 −

417. 14

987 − 418.

6

122 −

419. 24

721216 − 420.

12

6039 +

Section 8.6 Solving Equations with Radicals Objective 1 Solve radical equations having square root radicals. Solve the equation.

421. 9=w 422. 01 =−q

423. 68 =− y 424. 616 =+b

425. 462 =+q 426. 052 =−z

427. kk 232 =+ 428. rr 3156 =+

429. xx 333 =+ 430. 056 =−+t

431. 1665 −=+ tt 432. 153184 +=+ xx


Objective 2 Identify equations with no solutions. Solve the equation if it has a solution.

433. 10−=y 434. 08 =− p

435. 07 =+z 436. r−= 50

437. 13 −=+y 438. 075 =+−k

439. 062 =++x 440. 33132 +=+ mm

441. nn −=− 234 442. 6423 +=− pp

443. 1262 +−= rrr 444. 442 ++= sss

445. 1532 +−= bbb 446. 01242 =+++ ddd Objective 3 Solve equations by squaring a binomial. Find all solutions for the equation.

447. 115 +=+ yy 448. 35 +=+ mm

449. xx =+− 55 450. xx =−+ 113

451. 523 2 ++=+ ttt 452. 1181 2 ++=− qqq

453. 245 +=− pp 454. 113 +=− aa

455. 82 −=− xx 456. 121 +=+ yy

457. 42 −=− bb 458. 113 +=+ pp

459. 322 +=+ cc 460. 356 +=− tt

461. 9133 +=+ xx 462. 042 =+− xx


463. 132 =−−+ tt 464. 411 +=+− aa

465. 1273 +−=+ kk 466. xx +=+ 321

467. 13233 ++=+ xx 468. 5112 =++− rr Objective 4 Solve radical equations having cube root radicals. Solve each equation.

469. 3 33 1 5x x+ = 470. 3 32 3 5x x= +

471. 3 2 3 6 5x x= − 472. 3 2 3 4 4x x= −

473. 3 2 33 12 8 2x x x+ − = 474. 3 2 38 6 3 4x x x+ − = Mixed Exercises Find all solutions for the equation.

475. 58 =−z 476. 432 =+q

477. 073 =+r 478. 0276 =+−p

479. 1683 += rr 480. 1455 +=− yy

481. 2 2 1 2 1x x+ = + 482. 712 −=+ xx

483. nn 243 =+ 484. 56 =+m

485. 22 4 5 25t t t= − + 486. 12693 2 +−= ttt

487. 5322 −=+ xx 488. 1533 −=+ mm

489. 1232 −−= ppp 490. 2 2 10k k k= − +


491. 1043 −=− xx 492. 30542 2 −+= rrr

493. 12134 −=+ xx 494. 1176 +=−+ xx

495. 0357 =−−y 496. 33 −=− xx

497. 32157 +=+ yy 498. 444 =++− aa

499. 0513 =+−x 500. 3365 ++=+ cc

501. 283 −=− qq 502. 192210 +−=+ kk

503. 3 32 5 4x x+ = 504. 3 33 2 4x x= −

505. 3 2 3 5 6x x= − 506. 3 2 3 6 9x x= − −

507. 3 2 32 5 10x x x+ − = 508. 3 2 36 2 5 15x x x+ − = Writing/Conceptual Exercises 509. How can you tell that the equation 912 −=+x has no real number solution without

performing any algebraic steps? 510. Explain why the equation 812 =x has two real number solutions, while the equation

9=x has only one real number solution. 511. A student is told that he must check his solutions to the equation

42 −=− rr . The student said that he doesn’t see why this is necessary because he is sure he didn’t make any mistakes in solving the equation. How would you respond?

512. Explain why the equation

432 ++= zzz cannot have a negative solution.


Section 8.7 Using Rational Numbers as Exponents Objective 1 Define and use expressions of the form na /1 . Evaluate the expression. 513. 2/116 514. 2/125 515. 2/1169 516. 2/1225 517. 3/18 518. 3/127 519. 3/164 520. 3/1125 521. 3/1216 522. 3/1512 523. 3/1729 524. 3/11000 525. 4/116 526. 4/181 527. 4/1625 528. 4/11296 529. 5/132 530. 5/1243 531. 6/164 532. 7/1128 Objective 2 Define and use expressions of the form nma / . Evaluate the expression. 533. 2/316 534. 2/325 535. 2/336 536. 2/364 537. 3/28 538. 3/48 539. 3/227 540. 3/427 541. 3/264 542. 3/2125 543. 3/2216 544. 3/2512 545. 3/28− 546. 2/325− 547. 2/336− 548. 2/364− 549. 4/316− 550. 3/227− 551. 3/427− 552. 3/21000− Objective 3 Apply the rules for exponents using rational exponents. Simplify the expression. Write the answer in exponential form with only positive exponents. Assume that all variables represent positive numbers. 553. 2/32/1 55 ⋅ 554. 3/43/2 66 ⋅ 555. 8/38/5 77 ⋅ 556. 7/67/8 44 ⋅ 557. 8/38/5 88 −− ⋅ 558. 2/32/1 99 −− ⋅

559. 4/1

4/3

25

25 560.

2/1

2/1

8

8−

561. 5/3

5/4

6

6−


562. ( ) 2/33/24 sr 563. ( )33/29 564. ( )105/34

565. 2/1

8/5

12

12−

566. 2/3

9

4⎟⎠⎞

⎜⎝⎛ 567.

3/2

27

64−

⎟⎠⎞

⎜⎝⎛

568. 12

4/1

3/2

⎟⎟⎠

⎞⎜⎜⎝

⎛b

a 569.

3/2

2/1

2/1

⎟⎟⎠

⎞⎜⎜⎝

⎛−s

r 570.

2

2/32/1

y

yy −⋅

571. r

rr 4/14/3 −− ⋅ 572.

3/2

3/73/2

z

zz ⋅−

573. 3/2

3

6

⎟⎟⎠

⎞⎜⎜⎝

⎛x

c

574. 2

5/85/3

8

88−

−⋅ 575.

2

4/3

4/1

⎟⎟⎠

⎞⎜⎜⎝

⎛−x

x 576. ( ) ( ) 2/132/11 −−− aa

Objective 4 Use rational exponents to simplify radicals. Simplify the radical by first writing it in exponential form. Give the answer as an integer or a radical in simplest form. Assume that all variables represent nonnegative numbers.

577. 4 27 578. 6 33 579. 4 24

580. 6 39 581. 9 3125 582. 8 281

583. 6 2p 584. 6 3z 585. 6 2r

586. 8 2a 587. 8 216 588. 4 2100 Mixed Exercises Evaluate the expression.

589. 4/1625 590. 2/316− 591. 8 281

592. ( )63/23 593. 3/4

125

8⎟⎠⎞

⎜⎝⎛ 594. 3/427

595. 10/11024 596. 2/364− 597. 3/1

8

1⎟⎠⎞

⎜⎝⎛


598. 5/332 599. 9 627 600. 2/336

601. 5/1000,100 602. 9 664 603. 3/1343 604. 2/3100− 605. 7/97/5 99 ⋅ 606. 2/336− Simplify the expression. Write the answer in exponential form with only positive exponents. Assume that all variables represent positive numbers.

607. 2/32/1

2

yy

y−

−

608. 3/4

3/73/2

x

xx −⋅ 609.

24

3/2

4/3

⎟⎟⎠

⎞⎜⎜⎝

⎛y

x

610. 8/5

4/1

12

12−

611. ( ) 2/14/33/2 sr 612. 4/1

2/1

3/1

⎟⎟⎠

⎞⎜⎜⎝

⎛b

a

Writing/Conceptual Exercises Decide which one of the four choices is not equal to the given expression.

613. 2/1121 (a) 5.121 (b) 121 (c) 11− (d) 11

614. 5/232 (a) 4 (b) 22 (c) 5 1024 (d) 4

1

615. 3/11000− (a) 10− (b) 10

1 (c)

3 1000

1 (d) 0.1

616. 3/48− (a) 16− (b) 3 48− (c) 16

1 (d) 42−

chapter 8 roots and radicals - way cool …€¦ · · 2014-09-11chapter eight additional...

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