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Mathematics LibreTexts

5.E: Radical Functions and Equations (Exercises)

  • Anonymous
  • LibreTexts

( \newcommand{\kernel}{\mathrm{null}\,}\)

Exercise 5.E.1

Simplify.

  1. 121
  2. (7)2
  3. (xy)2
  4. (6x7)2
  5. 3125
  6. 327
  7. 3(xy)3
  8. 3(6x+1)3
  9. Given f(x)=x+10, find f(1) and f(6).
  10. Given g(x)=3x5, find g(4) and g(13).
  11. Determine the domain of the function defined by g(x)=5x+2.
  12. Determine the domain of the function defined by g(x)=33x1.
Answer

1. 11

3. |xy|

5. 5

7. xy

9. f(1)=3;f(6)=4

11. [25,)

Exercise 5.E.2

Simplify.

  1. 3250
  2. 43120
  3. 33108
  4. 105132
  5. 648116
  6. 6128
  7. 5192
  8. 3420
Answer

1. 532

3. 934

5. 9

7. 256

Exercise 5.E.3

Simplify.

  1. 20x4y3
  2. 454x6y3
  3. x214x+49
  4. (x8)4
Answer

1. 2x2|y|5y

3. |x7|

Exercise 5.E.4

Simplify. (Assume all variable expressions are nonzero.)

  1. 100x2y4
  2. 36a6b2
  3. 8a2b4
  4. 72x4yz6
  5. 10x150x7y4
  6. 5n225m10n6
  7. 348x6y3z2
  8. 3270a10b8c3
  9. 3a3b564c6
  10. 5a2632b5c10
  11. The period T in seconds of a pendulum is given by the formula T=2πL32 where L represents the length in feet of the pendulum. Calculate the period of a pendulum that is 212 feet long. Give the exact answer and the approximate answer to the nearest hundredth of a second.
  12. The time in seconds an object is in free fall is given by the formula t=s4 where s represents the distance in feet the object has fallen. How long does it take an object to fall 28 feet? Give the exact answer and the approximate answer to the nearest tenth of a second.
  13. Find the distance between (5,6) and (3,4).
  14. Find the distance between (23,12) and (1,34).
Answer

1. 10xy2

3. 2a2b2

5. 50x4y26x

7. 2x2y36z2

9. ab3b24c2

11. π54 seconds; 1.76 seconds

13. 226 units

Exercise 5.E.5

Determine whether or not the three points form a right triangle. Use the Pythagorean theorem to justify your answer.

  1. (4,5),(3,1), and (3,0)
  2. (1,1),(1,3), and (6,1)
Answer

1. Right triangle

Exercise 5.E.6

Simplify. Assume all radicands containing variables are nonnegative.

  1. 72+52
  2. 815215
  3. 143+525362
  4. 22ab5ab+7ab2ab
  5. 7x(3x+2y)
  6. (8yx7xy)(5xy12yx)
  7. (35+26)+(8536)
  8. (433312)(5332312)
  9. (210x+3y)(1+210x6y)
  10. (3a3ab2+63a2b)+(9a3ab2123a2b)
  11. 45+122075
  12. 2432+54232
  13. 23x2+45xx27+20x
  14. 56a2b+8a2b2224a2ba18b2
  15. 5y4x2y(x16y329x2y3)
  16. (2b9a2c3a16b2c)(64a2b2c9ba2c)
  17. 3216x3125xy38x
  18. 3128x32x354+332x3
  19. 38x3y2x38y+327x3y+x3y
  20. 327a3b338ab3+a364bb3a
  21. Calculate the perimeter of the triangle formed by the following set of vertices: {(3,2),(1,1),(1,2)}.
  22. Calculate the perimeter of the triangle formed by the following set of vertices: {(0,4),(2,0),(3,0)}.
Answer

1. 122

3. 932

5. 4x2y

7. 1156

9. 1310x+9y

11. 533

13. x3+55x

15. 12xyy

17. 43x53xy

19. 2x3y

21. 4+213 units

Exercise 5.E.7

Multiply.

  1. 615
  2. (42)2
  3. 2(210)
  4. (56)2
  5. (53)(5+3)
  6. (26+3)(253)
  7. (a5b)2
  8. 3xy(x2y)
  9. 33a2318a
  10. 349a2b37a2b2
Answer

1. 310

3. 225

5. 22

7. a10ab+25b

9. 3a32

Exercise 5.E.8

Divide. Assume all variables represent nonzero numbers and rationalize the denominator where appropriate.

  1. 729
  2. 104864
  3. 55
  4. 152
  5. 326
  6. 2+510
  7. 183x
  8. 23x6xy
  9. 133x2
  10. 5ab235a2b
  11. 35xz249x2y2z
  12. 158x4y2z
  13. 9x2y581xy2z3
  14. 527ab315a4bc2
  15. 153
  16. 32+1
  17. 36210
  18. xyxy
  19. 262+6
  20. a+bab
  21. The base of a triangle measures 26 units and the height measures 315 units. Find the area of the triangle.
  22. If each side of a square measures 5+210 units, find the area of the square.
Answer

1. 22

3. 5

5. 64

7. 63xx

9. 39x3x

11. 335x2yz7xy

13. 3xy53x4y3z2z

15. 5+32

17. 6+15

19. 2+3

21. 910 square units

Exercise 5.E.9

Express in radical form.

  1. 111/2
  2. 22/3
  3. x3/5
  4. a4/5
Answer

1. 11

3. 5x3

Exercise 5.E.10

Write as a radical and then simplify.

  1. 161/2
  2. 721/2
  3. 82/3
  4. 321/3
  5. (19)3/2
  6. (1216)1/3
Answer

1. 4

3. 4

5. 127

Exercise 5.E.11

Perform the operations and simplify. Leave answers in exponential form.

  1. 61/263/2
  2. 31/331/2
  3. 65/263/2
  4. 43/441/4
  5. (64x6y2)1/2
  6. (27x12y6)1/3
  7. (a4/3a1/2)2/5
  8. (16x4/3y2)1/2
  9. 56x3/4y3/214x1/2y2/3
  10. (4a4b2/3c4/3)1/22a2b1/6c2/3
  11. (9x4/3y1/3)3/2
  12. (16x4/5y1/2z2/3)3/4
Answer

1. 36

3. 6

5. 8x3y

7. a1/3

9. 4x1/4y5/6

11. x227y1/2

Exercise 5.E.12

Perform the operations with mixed indices.

  1. y5y2
  2. 3y5y3
  3. 3y23y
  4. 3y2
Answer

1. 10y9

3. 15y7

Exercise 5.E.13

Solve.

  1. 2x+3=13
  2. 3x2=4
  3. x5+4=8
  4. 5x+3+7=2
  5. 4x3=2x+15
  6. 8x15=x
  7. x1=13x
  8. 4x3=2x3
  9. x+5=5x
  10. x+3=3x1
  11. 2(x+1)x+2=1
  12. 6x+x2=2
  13. 3x2+x1=1
  14. 9x=x+161
  15. 34x3=2
  16. 3x8=1
  17. 3x(3x+10)=2
  18. 32x2x+4=5
  19. 33(x+4)(x+1)=35x+37
  20. 33x29x+24=3(x+2)2
  21. y1/23=0
  22. y1/3+3=0
  23. (x5)1/22=0
  24. (2x1)1/35=0
  25. (x1)1/2=x1/21
  26. (x2)1/2(x6)1/2=2
  27. (x+4)1/2(3x)1/2=2
  28. (5x+6)1/2=3(x+3)1/2
  29. Solve for g:t=2sg.
  30. Solve for x:y=3x+42,
  31. The period in seconds of a pendulum is given by the formula T=2πL32 where L represents the length in feet of the pendulum. Find the length of a pendulum that has a period of 112 seconds. Find the exact answer and the approximate answer rounded off to the nearest tenth of a foot.
  32. The outer radius of a spherical shell is given by the formula r=33V4π+2 where V represents the inner volume in cubic centimeters. If the outer radius measures 8 centimeters, find the inner volume of the sphere.
  33. The speed of a vehicle before the brakes are applied can be estimated by the length of the skid marks left on the road. On dry pavement, the speed v in miles per hour can be estimated by the formula v=26d, where d represents the length of the skid marks in feet. Estimate the length of a skid mark if the vehicle is traveling 30 miles per hour before the brakes are applied.
  34. Find the real root of the function defined by f(x)=3x3+2.
Answer

1. 25

3. 21

5. 9

7. 4

9. 4

11. 7

13. 1

15. 114

17. 4,23

19. 5,53

21. 9

23. 9

25. 1

27. 12

29. g=2st2

31. 18π2 feet; 1.8 feet

33. 37.5 feet

Exercise 5.E.14

Write the complex number in standard form a+bi.

  1. 516
  2. 256
  3. 3+810
  4. 1246
Answer

1. 54i

3. 310+25i

Exercise 5.E.15

Perform the operations.

  1. (612i)+(4+7i)
  2. (3+2i)(64i)
  3. (12i)(3432i)
  4. (5815i)+(3223i)
  5. (52i)(67i)+(44i)
  6. (103i)+(20+5i)(3015i)
  7. 4i(23i)
  8. (2+3i)(52i)
  9. (4+i)2
  10. (83i)2
  11. (3+2i)(32i)
  12. (1+5i)(15i)
  13. 2+9i2i
  14. i12i
  15. 4+5i2i
  16. 32i3+2i
  17. 105(23i)2
  18. (23i)2(23i)+4
  19. (11i)2
  20. (1+2i3i)2
  21. 8(34)
  22. (118)(32)
  23. (510)2
  24. (12)2(1+2)2
  25. Show that both 5i and 5i satisfy x2+25=0.
  26. Show that both 12i and 1+2i satisfy x22x+5=0.
Answer

1. 105i

3. 14+12i

5. 3+i

7. 12+8i

9. 15+8i

11. 13

13. 92i

15. 35+145i

17. 35+60i

19. 12i

21. 42+2i6

23. 15+102

25. Answer may vary

Sample Exam

Exercise 5.E.16

Simplify. (Assume all variables are positive.)

  1. 5x121x2y4
  2. 2xy2364x6y9
  3. Calculate the distance between (5,3) and (2,6).
  4. The time in seconds an object is in free fall is given by the formula t=s4 where s represents the distance in feet that the object has fallen. If a stone is dropped into a 36-foot pit, how long will it take to hit the bottom of the pit?
Answer

1. 55x2y2

3. 310 units

Exercise 5.E.17

Perform the operations and simplify. (Assume all variables are positive and rationalize the denominator where appropriate.)

  1. 150xy2218x3+y24x+x128x
  2. 3316x3y2(2x3250y2354x3y2)
  3. 22(236)
  4. (105)2
  5. 62+3
  6. 2x2xy
  7. 158xy2z4
  8. Simplify: 813/4.
  9. Express in radical form: x3/5.
Answer

1. 7y6x+2x2x

3. 4123

5. 23+32

7. 54x4y3z2xyz

9. 15x3

Exercise 5.E.18

Solve.

  1. x5=1
  2. 35x2+6=4
  3. 52x+52x=11
  4. 43x+2=x
  5. 2x+5x+3=2
  6. The time in seconds an object is in free fall is given by the formula t=s4 where s represents the distance in feet that the object has fallen. If a stone is dropped into a pit and it takes 4 seconds to reach the bottom, how deep is the pit?
  7. The width in inches of a container is given by the formula w=34V2+1 where V represents the inside volume in cubic inches of the container. What is the inside volume of the container if the width is 6 inches?
Answer

2. 65

4.

6. 256 feet

Exercise 5.E.19

Perform the operations and write the answer in standard form.

  1. 3(63)
  2. 4+3i2i
  3. 63(23i)2
Answer

1. 3+3i2

3. 21+36i


This page titled 5.E: Radical Functions and Equations (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Anonymous via source content that was edited to the style and standards of the LibreTexts platform.

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