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5.8 Review Exercises and Sample Exam

Review Exercises

    Roots and Radicals

      Simplify.

    1. 121

    2. (7)2

    3. (xy)2

    4. (6x7)2

    5. 1253

    6. 273

    7. (xy)33

    8. (6x+1)33

    9. Given f(x)=x+10, find f(1) and f(6).

    10. Given g(x)=x53, 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)=3x13.

      Simplify.

    1. 2503

    2. 41203

    3. 31083

    4. 101325
    5. 681164
    6. 1286

    7. 1925

    8. 3420

    Simplifying Radical Expressions

      Simplify.

    1. 20x4y3

    2. 454x6y3

    3. x214x+49

    4. (x8)4

      Simplify. (Assume all variable expressions are nonzero.)

    1. 100x2y4

    2. 36a6b2

    3. 8a2b4
    4. 72x4yz6
    5. 10x150x7y4

    6. 5n225m10n6

    7. 48x6y3z23

    8. 270a10b8c33

    9. a3b564c63
    10. a2632b5c105
    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).

      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)

    Adding and Subtracting Radical Expressions

      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. (433123)(5332123)

    9. (210x+3y)(1+210x6y)

    10. (3aab23+6a2b3)+(9aab2312a2b3)
    11. 45+122075

    12. 2432+54232

    13. 23x2+45xx27+20x

    14. 56a2b+8a2b2224a2ba18b2

    15. 5y4x2y(x16y329x2y3)

    16. (2b9a2c3a16b2c)(64a2b2c9ba2c)

    17. 216x3125xy38x3

    18. 128x332x543+32x33

    19. 8x3y32x8y3+27x3y3+xy3

    20. 27a3b338ab33+a64b3ba3

    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)}.

    Multiplying and Dividing Radical Expressions

      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. 3a2318a3

    10. 49a2b37a2b23

      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. 13x23
    10. 5ab25a2b3
    11. 5xz249x2y2z3
    12. 18x4y2z5
    13. 9x2y81xy2z35
    14. 27ab315a4bc25
    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.

    Rational Exponents

      Express in radical form.

    1. 111/2

    2. 22/3

    3. x3/5

    4. a4/5

      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

      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

      Perform the operations with mixed indices.

    1. yy25

    2. y3y35

    3. y23y5

    4. y23

    Solving Radical Equations

      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. 4x33=2

    16. x83=1

    17. x(3x+10)3=2

    18. 2x2x3+4=5

    19. 3(x+4)(x+1)3=5x+373

    20. 3x29x+243=(x+2)23

    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=x+432.

    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=3V4π3+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)=x33+2.

    Complex Numbers and Their Operations

      Write the complex number in standard form a+bi.

    1. 516

    2. 256

    3. 3+810
    4. 1246

      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.

Answers

  1. −11

  2. |xy|

  3. 5

  4. xy

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

  6. [25,)
  7. 523

  8. 943

  9. −9

  10. 265

  1. 2x2|y|5y

  2. |x7|

  3. 10xy2

  4. 2a2b2
  5. 50x4y26x

  6. 2x2y6z23

  7. abb234c2
  8. π54 seconds; 1.76 seconds

  9. 226 units

  10. Right triangle

  1. 122

  2. 932

  3. 4x2y

  4. 1156

  5. 1310x+9y

  6. 533

  7. x3+55x

  8. 12xyy

  9. 4x35xy3

  10. 2xy3

  11. 4+213 units

  1. 310

  2. 225

  3. 22

  4. a10ab+25b

  5. 3a23

  6. 22

  7. 5

  8. 64
  9. 63xx
  10. 9x33x
  11. 35x2yz37xy
  12. 3xy3x4y3z25z
  13. 5+32
  14. 6+15

  15. 2+3

  16. 910 square units

  1. 11

  2. x35

  3. 4

  4. 4

  5. 1/27

  6. 36

  7. 6

  8. 8x3y

  9. a1/3

  10. 4x1/4y5/6

  11. x227y1/2
  12. y910

  13. y715

  1. 25

  2. 21

  3. 9

  4. 4

  5. 4

  6. 7

  7. 1

  8. 114

  9. 4,23

  10. 5,53

  11. 9

  12. 9

  13. 1

  14. 12

  15. g=2st2

  16. 18π2 feet; 1.8 feet

  17. 37.5 feet

  1. 54i

  2. 310+25i
  3. 105i

  4. 14+12i

  5. 3+i

  6. 12+8i

  7. 15+8i

  8. 13

  9. 92i

  10. 35+145i

  11. 35+60i

  12. 12i

  13. 42+2i6

  14. 15+102

  15. Answer may vary

Sample Exam

      Simplify. (Assume all variables are positive.)

    1. 5x121x2y4

    2. 2xy264x6y93

    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?

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

    1. 150xy2218x3+y24x+x128x

    2. 316x3y23(2x250y2354x3y23)
    3. 22(236)

    4. (105)2

    5. 62+3
    6. 2x2xy
    7. 18xy2z45
    8. Simplify: 813/4.

    9. Express in radical form: x3/5.

      Simplify. Assume all variables are nonzero and leave answers in exponential form.

    1. (81x4y2)1/2

    2. (25a4/3b8)3/2a1/2b

      Solve.

    1. x5=1

    2. 5x23+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=4V32+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?

      Perform the operations and write the answer in standard form.

    1. 3(63)

    2. 4+3i2i

    3. 63(23i)2

Answers

  1. 55x2y2

  2. 310 units

  3. 7y6x+2x2x

  4. 4123

  5. 23+32

  6. 4x4y3z52xyz
  7. 1x35
  8. 125a3/2b11

  9. 65

  10. Ø

  11. 256 feet

  12. 3+3i2

  13. 21+36i