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Cubes and Cube Roots

Question paper, Set 5

  • Set 5 of 10
  • 20 questions
  • 50 marks
  • 1 hour 30 minutes

Download the PDF (with key sheet and solutions, 73.7 KB)

Work each question on paper first, then open its answer and solution to check your work.

  1. Express 13×13×1313 \times 13 \times 13 as a cube.
    1 mark
    Answer and solution

    1313 is multiplied by itself 3 times.

    ∴13×13×13=133\therefore 13 \times 13 \times 13 = 13^3

    Answer: 13313^3

  2. Expand 636^3 and find its value.
    2 marks
    Answer and solution

    63=6×6×66^3 = 6 \times 6 \times 6

    =36×6=216= 36 \times 6 = 216

    Answer: 6×6×6=2166 \times 6 \times 6 = 216

  3. Fill in the blank: □3=343\square^3 = 343
    1 mark
    Answer and solution

    7×7×7=3437 \times 7 \times 7 = 343

    ∴343=73\therefore 343 = 7^3

    Answer: 77

  4. Find 434^3 and 737^3.
    2 marks
    Answer and solution

    43=4×4×4=644^3 = 4 \times 4 \times 4 = 64

    73=7×7×7=3437^3 = 7 \times 7 \times 7 = 343

    Answer: 43=644^3 = 64, 73=3437^3 = 343

  5. Write the cubes of all the odd numbers from 1 to 10.
    3 marks
    Answer and solution

    13=1×1×1=11^3 = 1 \times 1 \times 1 = 1

    33=3×3×3=273^3 = 3 \times 3 \times 3 = 27

    53=5×5×5=1255^3 = 5 \times 5 \times 5 = 125

    73=7×7×7=3437^3 = 7 \times 7 \times 7 = 343

    93=9×9×9=7299^3 = 9 \times 9 \times 9 = 729

    Answer: 13=11^3 = 1, 33=273^3 = 27, 53=1255^3 = 125, 73=3437^3 = 343, 93=7299^3 = 729

  6. Find the cube of 50.
    2 marks
    Answer and solution

    50×50=250050 \times 50 = 2500

    2500×50=1250002500 \times 50 = 125000

    ∴503=125000\therefore 50^3 = 125000

    Answer: 125000125000

  7. Find 12312^3, 30330^3 and 17317^3. Multiply each number by itself first, then by the number again.
    5 marks
    Answer and solution

    12×12=14412 \times 12 = 144, 144×12=1728144 \times 12 = 1728

    30×30=90030 \times 30 = 900, 900×30=27000900 \times 30 = 27000

    17×17=28917 \times 17 = 289, 289×17=4913289 \times 17 = 4913

    123=1728,303=27000,173=491312^3 = 1728, 30^3 = 27000, 17^3 = 4913

    Answer: 123=172812^3 = 1728, 303=2700030^3 = 27000, 173=491317^3 = 4913

  8. Simplify (−y)3(-y)^3.
    1 mark
    Answer and solution

    (−y)3=(−y)×(−y)×(−y)(-y)^3 = (-y) \times (-y) \times (-y)

    =−y3= -y^3

    Answer: −y3-y^3

  9. Show that (−7)3=−73(-7)^3 = -7^3.
    2 marks
    Answer and solution

    LHS =(−7)3=(−7)×(−7)×(−7)=49×(−7)=−343= (-7)^3 = (-7) \times (-7) \times (-7) = 49 \times (-7) = -343

    RHS =−73=−(7×7×7)=−343= -7^3 = -(7 \times 7 \times 7) = -343

    LHS = RHS, so (−7)3=−73(-7)^3 = -7^3.

    Answer: Both sides are −343-343

  10. Find (−9)3(-9)^3 and 939^3. Which of the two is bigger, and why?
    3 marks
    Answer and solution

    93=9×9×9=7299^3 = 9 \times 9 \times 9 = 729

    (−9)3=(−9)×(−9)×(−9)=−729(-9)^3 = (-9) \times (-9) \times (-9) = -729

    729729 is positive and −729-729 is negative, so 939^3 is bigger.

    Answer: 93=7299^3 = 729 is bigger than (−9)3=−729(-9)^3 = -729

  11. Find the value of 13+631^3 + 6^3.
    1 mark
    Answer and solution

    13+63=1+2161^3 + 6^3 = 1 + 216

    =217= 217

    Answer: 217217

  12. Find 43+53+63+734^3 + 5^3 + 6^3 + 7^3.
    5 marks
    Answer and solution

    43+53+63+73=64+125+216+3434^3 + 5^3 + 6^3 + 7^3 = 64 + 125 + 216 + 343

    =748= 748

    Answer: 748748

  13. Find the value of (−x)3+x3(-x)^3 + x^3.
    1 mark
    Answer and solution

    (−x)3=−x3(-x)^3 = -x^3

    ∴x3+(−x)3=x3−x3=0\therefore x^3 + (-x)^3 = x^3 - x^3 = 0

    Answer: 00

  14. Find (−2)3+43+(−3)3(-2)^3 + 4^3 + (-3)^3.
    2 marks
    Answer and solution

    (−2)3+43+(−3)3=−8+64+(−27)(-2)^3 + 4^3 + (-3)^3 = -8 + 64 + (-27)

    =−8+64−27= -8 + 64 - 27

    =29= 29

    Answer: 2929

  15. Find the value of 23−932^3 - 9^3.
    2 marks
    Answer and solution

    23−93=8−7292^3 - 9^3 = 8 - 729

    939^3 is bigger than 232^3, so the answer is negative.

    =−721= -721

    Answer: −721-721

  16. Find 73−337^3 - 3^3 and 33−733^3 - 7^3. Which answer is negative, and why?
    3 marks
    Answer and solution

    73−33=343−27=3167^3 - 3^3 = 343 - 27 = 316

    33−73=27−343=−3163^3 - 7^3 = 27 - 343 = -316

    In the second, the bigger cube 343343 is taken from the smaller cube 2727, so the answer is negative.

    Answer: 316316 and −316-316; the second is negative

  17. Sita packs dice into two cube-shaped boxes. The first box holds 6 dice along each edge and the second holds 9. (i) How many dice does each box hold? (ii) How many more dice does the second box hold?
    5 marks
    Answer and solution

    (i) First: 63=6×6×6=2166^3 = 6 \times 6 \times 6 = 216

    Second: 93=9×9×9=7299^3 = 9 \times 9 \times 9 = 729

    (ii) 729−216=513729 - 216 = 513

    Answer: (i) 216216 and 729729 (ii) 513513

  18. Find 23−(−2)32^3 - (-2)^3.
    1 mark
    Answer and solution

    23−(−2)3=8−(−8)2^3 - (-2)^3 = 8 - (-8)

    =8+8= 8 + 8

    =16= 16

    Answer: 1616

  19. Find 33×(−4)33^3 \times (-4)^3.
    3 marks
    Answer and solution

    33=273^3 = 27, (−4)3=−64(-4)^3 = -64

    27×(−64)=−172827 \times (-64) = -1728

    Answer: −1728-1728

  20. Find (−1)3+(−2)3+(−3)3+(−4)3+(−5)3+(−6)3(-1)^3 + (-2)^3 + (-3)^3 + (-4)^3 + (-5)^3 + (-6)^3.
    5 marks
    Answer and solution

    (−1)3+(−2)3+(−3)3+(−4)3+(−5)3+(−6)3=−1+(−8)+(−27)+(−64)+(−125)+(−216)(-1)^3 + (-2)^3 + (-3)^3 + (-4)^3 + (-5)^3 + (-6)^3 = -1 + (-8) + (-27) + (-64) + (-125) + (-216)

    =−1−8−27−64−125−216= -1 - 8 - 27 - 64 - 125 - 216

    =−441= -441

    Answer: −441-441