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Cubes and Cube Roots

Question paper, Set 8

  • Set 8 of 10
  • 20 questions
  • 50 marks
  • 1 hour 30 minutes

Download the PDF (with key sheet and solutions, 74.1 KB)

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

  1. Write z3z^3 as a product of zz's.
    1 mark
    Answer and solution

    z3z^3 means zz multiplied by itself 3 times.

    ∴z3=z×z×z\therefore z^3 = z \times z \times z

    Answer: z×z×zz \times z \times z

  2. Ravi says 73=7×3=217^3 = 7 \times 3 = 21. Is he right? Give the correct value.
    2 marks
    Answer and solution

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

    7×3=217 \times 3 = 21

    737^3 means 7 multiplied by itself 3 times, not 7 times 3.

    Answer: No. 73=3437^3 = 343, but 7×3=217 \times 3 = 21

  3. 27 is the cube of which number?
    1 mark
    Answer and solution

    3×3×3=273 \times 3 \times 3 = 27

    ∴27=33\therefore 27 = 3^3

    Answer: 33

  4. A carton holds soap boxes in 7 layers. Each layer has 7 rows with 7 boxes in each row. How many soap boxes are in the carton?
    2 marks
    Answer and solution

    Number = 7×7×77 \times 7 \times 7

    =73=49×7=343= 7^3 = 49 \times 7 = 343

    Answer: 343343 soap boxes

  5. Fill in the boxes using the table of cubes: (i) 73=□7^3 = \square (ii) □3=216\square^3 = 216 (iii) 103=□10^3 = \square
    3 marks
    Answer and solution

    (i) 73=7×7×7=3437^3 = 7 \times 7 \times 7 = 343

    (ii) 6×6×6=2166 \times 6 \times 6 = 216, so the box is 6.

    (iii) 103=10×10×10=100010^3 = 10 \times 10 \times 10 = 1000

    Answer: (i) 343343 (ii) 66 (iii) 10001000

  6. Find the value of 90×90×9090 \times 90 \times 90.
    2 marks
    Answer and solution

    90×90=810090 \times 90 = 8100

    8100×90=7290008100 \times 90 = 729000

    ∴903=729000\therefore 90^3 = 729000

    Answer: 729000729000

  7. Find 434^3 and 40340^3. 40340^3 is how many times 434^3?
    5 marks
    Answer and solution

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

    40×40=160040 \times 40 = 1600, 1600×40=640001600 \times 40 = 64000

    64000÷64=100064000 \div 64 = 1000

    So 40340^3 is 1000 times 434^3.

    Answer: 43=644^3 = 64, 403=6400040^3 = 64000; 40340^3 is 1000 times 434^3

  8. Write the value of (−6)3(-6)^3.
    1 mark
    Answer and solution

    (−6)3=−63(-6)^3 = -6^3

    =−216= -216

    Answer: −216-216

  9. Find (−8)3(-8)^3 and (−20)3(-20)^3.
    2 marks
    Answer and solution

    (−8)3=−83=−512(-8)^3 = -8^3 = -512

    20×20=40020 \times 20 = 400, 400×20=8000400 \times 20 = 8000

    ∴(−20)3=−8000\therefore (-20)^3 = -8000

    Answer: (−8)3=−512(-8)^3 = -512, (−20)3=−8000(-20)^3 = -8000

  10. Find 838^3 and (−8)3(-8)^3. How are the two answers related?
    3 marks
    Answer and solution

    83=8×8×8=5128^3 = 8 \times 8 \times 8 = 512

    (−8)3=(−8)×(−8)×(−8)=−512(-8)^3 = (-8) \times (-8) \times (-8) = -512

    The two answers are the same number with opposite signs: (−8)3=−83(-8)^3 = -8^3.

    Answer: 83=5128^3 = 512, (−8)3=−512(-8)^3 = -512; same number, opposite signs

  11. Add the cube of 9 to itself.
    1 mark
    Answer and solution

    93+93=729+7299^3 + 9^3 = 729 + 729

    =1458= 1458

    Answer: 14581458

  12. Find 13+23+33+⋯+731^3 + 2^3 + 3^3 + \cdots + 7^3.
    5 marks
    Answer and solution

    13+23+33+43+53+63+73=1+8+27+64+125+216+3431^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 = 1 + 8 + 27 + 64 + 125 + 216 + 343

    =784= 784

    Answer: 784784

  13. Simplify p3+(−p)3p^3 + (-p)^3.
    1 mark
    Answer and solution

    (−p)3=−p3(-p)^3 = -p^3

    ∴p3+(−p)3=p3−p3=0\therefore p^3 + (-p)^3 = p^3 - p^3 = 0

    Answer: 00

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

    (−3)3+(−4)3=−27+(−64)(-3)^3 + (-4)^3 = -27 + (-64)

    =−27−64= -27 - 64

    =−91= -91

    Answer: −91-91

  15. What must be added to 535^3 to get 838^3?
    2 marks
    Answer and solution

    Number to be added = 83−538^3 - 5^3

    83−53=512−1258^3 - 5^3 = 512 - 125

    =387= 387

    Answer: 387387

  16. What must be subtracted from 12312^3 to get 939^3?
    3 marks
    Answer and solution

    Number = 123−9312^3 - 9^3

    123=144×12=172812^3 = 144 \times 12 = 1728

    123−93=1728−72912^3 - 9^3 = 1728 - 729

    =999= 999

    Answer: 999999

  17. How much more is 10310^3 than 63+736^3 + 7^3?
    5 marks
    Answer and solution

    103=100010^3 = 1000

    63+73=216+343=5596^3 + 7^3 = 216 + 343 = 559

    Difference = 1000−559=4411000 - 559 = 441

    Answer: 441441

  18. Simplify −b3−(−b)3-b^3 - (-b)^3.
    1 mark
    Answer and solution

    (−b)3=−b3(-b)^3 = -b^3

    −b3−(−b3)=−b3+b3=0-b^3 - (-b^3) = -b^3 + b^3 = 0

    Answer: 00

  19. Find the value of 23−(−6)32^3 - (-6)^3.
    3 marks
    Answer and solution

    23−(−6)3=8−(−216)2^3 - (-6)^3 = 8 - (-216)

    =8+216= 8 + 216

    =224= 224

    Answer: 224224

  20. If a=−2a = -2, b=6b = 6 and c=−3c = -3, find a3−b3+c3a^3 - b^3 + c^3.
    5 marks
    Answer and solution

    a3−b3+c3=(−2)3−63+(−3)3a^3 - b^3 + c^3 = (-2)^3 - 6^3 + (-3)^3

    (−2)3−63+(−3)3=−8−216+(−27)(-2)^3 - 6^3 + (-3)^3 = -8 - 216 + (-27)

    =−8−216−27= -8 - 216 - 27

    =−251= -251

    Answer: −251-251