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Cubes and Cube Roots

Question paper, Set 9

  • Set 9 of 10
  • 20 questions
  • 50 marks
  • 1 hour 30 minutes

Download the PDF (with key sheet and solutions, 74.7 KB)

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

  1. Write 8×8×88 \times 8 \times 8 using the power 3.
    1 mark
    Answer and solution

    88 is multiplied by itself 3 times.

    ∴8×8×8=83\therefore 8 \times 8 \times 8 = 8^3

    Answer: 838^3

  2. Write 4×4×4×9×9×94 \times 4 \times 4 \times 9 \times 9 \times 9 using cubes.
    2 marks
    Answer and solution

    4×4×4=434 \times 4 \times 4 = 4^3

    9×9×9=939 \times 9 \times 9 = 9^3

    ∴4×4×4×9×9×9=43×93\therefore 4 \times 4 \times 4 \times 9 \times 9 \times 9 = 4^3 \times 9^3

    Answer: 43×934^3 \times 9^3

  3. Write the next number in the pattern 1000,729,512,343,…1000, 729, 512, 343, \ldots
    1 mark
    Answer and solution

    These are 10310^3, 939^3, 838^3, 737^3.

    ∴\therefore the next number is 63=2166^3 = 216

    Answer: 216216

  4. Using the table of cubes, write the values of 535^3 and 10310^3.
    2 marks
    Answer and solution

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

    103=10×10×10=100010^3 = 10 \times 10 \times 10 = 1000

    Answer: 53=1255^3 = 125, 103=100010^3 = 1000

  5. Write the cubes of 6, 7, 8, 9 and 10.
    3 marks
    Answer and solution

    63=6×6×6=2166^3 = 6 \times 6 \times 6 = 216

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

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

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

    103=10×10×10=100010^3 = 10 \times 10 \times 10 = 1000

    Answer: 63=2166^3 = 216, 73=3437^3 = 343, 83=5128^3 = 512, 93=7299^3 = 729, 103=100010^3 = 1000

  6. Fill in the boxes: 123=12×12×12=□×12=□12^3 = 12 \times 12 \times 12 = \square \times 12 = \square
    2 marks
    Answer and solution

    12×12=14412 \times 12 = 144

    144×12=1728144 \times 12 = 1728

    ∴123=144×12=1728\therefore 12^3 = 144 \times 12 = 1728

    Answer: 144144 and 17281728

  7. Find the values of 13313^3, 40340^3 and 16316^3, showing your working.
    5 marks
    Answer and solution

    13×13=16913 \times 13 = 169, 169×13=2197169 \times 13 = 2197

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

    16×16=25616 \times 16 = 256, 256×16=4096256 \times 16 = 4096

    133=2197,403=64000,163=409613^3 = 2197, 40^3 = 64000, 16^3 = 4096

    Answer: 133=219713^3 = 2197, 403=6400040^3 = 64000, 163=409616^3 = 4096

  8. Fill in the box: (−c)3=□(-c)^3 = \square
    1 mark
    Answer and solution

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

    =−c3= -c^3

    Answer: −c3-c^3

  9. Show that (−6)3=−63(-6)^3 = -6^3.
    2 marks
    Answer and solution

    LHS =(−6)3=(−6)×(−6)×(−6)=36×(−6)=−216= (-6)^3 = (-6) \times (-6) \times (-6) = 36 \times (-6) = -216

    RHS =−63=−(6×6×6)=−216= -6^3 = -(6 \times 6 \times 6) = -216

    LHS = RHS, so (−6)3=−63(-6)^3 = -6^3.

    Answer: Both sides are −216-216

  10. Find (−15)3(-15)^3 by first finding (−15)×(−15)(-15) \times (-15).
    3 marks
    Answer and solution

    (−15)×(−15)=225(-15) \times (-15) = 225

    225×(−15)=−3375225 \times (-15) = -3375

    ∴(−15)3=−3375\therefore (-15)^3 = -3375

    Answer: −3375-3375

  11. Find the sum of the cubes of 1 and 9.
    1 mark
    Answer and solution

    13+93=1+7291^3 + 9^3 = 1 + 729

    =730= 730

    Answer: 730730

  12. Find the sum of the cubes of all the odd numbers from 1 to 11.
    5 marks
    Answer and solution

    113=133111^3 = 1331

    13+33+53+73+93+113=1+27+125+343+729+13311^3 + 3^3 + 5^3 + 7^3 + 9^3 + 11^3 = 1 + 27 + 125 + 343 + 729 + 1331

    =2556= 2556

    Answer: 25562556

  13. Find 53+(−3)35^3 + (-3)^3.
    1 mark
    Answer and solution

    53+(−3)3=125+(−27)5^3 + (-3)^3 = 125 + (-27)

    =125−27= 125 - 27

    =98= 98

    Answer: 9898

  14. Find (−2)3+93(-2)^3 + 9^3.
    2 marks
    Answer and solution

    (−2)3+93=−8+729(-2)^3 + 9^3 = -8 + 729

    =721= 721

    Answer: 721721

  15. Find 43−1034^3 - 10^3.
    2 marks
    Answer and solution

    43−103=64−10004^3 - 10^3 = 64 - 1000

    10310^3 is bigger than 434^3, so the answer is negative.

    =−936= -936

    Answer: −936-936

  16. Find 63−936^3 - 9^3 and 93−639^3 - 6^3. What do you notice?
    3 marks
    Answer and solution

    63−93=216−729=−5136^3 - 9^3 = 216 - 729 = -513

    93−63=729−216=5139^3 - 6^3 = 729 - 216 = 513

    Changing the order of subtraction changes only the sign of the answer.

    Answer: −513-513 and 513513; same number with opposite signs

  17. Two cube-shaped cartons are packed with soap boxes, 5 along each edge in the first and 8 in the second. (i) How many soap boxes are in each carton? (ii) How many more soap boxes are in the second carton?
    5 marks
    Answer and solution

    (i) First: 53=5×5×5=1255^3 = 5 \times 5 \times 5 = 125

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

    (ii) 512−125=387512 - 125 = 387

    Answer: (i) 125125 and 512512 (ii) 387387

  18. Find (−1)3×73(-1)^3 \times 7^3.
    1 mark
    Answer and solution

    (−1)3=−1(-1)^3 = -1, 73=3437^3 = 343

    −1×343=−343-1 \times 343 = -343

    Answer: −343-343

  19. Find the value of −23−43-2^3 - 4^3.
    3 marks
    Answer and solution

    −23−43=−8−64-2^3 - 4^3 = -8 - 64

    =−72= -72

    Answer: −72-72

  20. Find the values of: (i) (−8)3−83(-8)^3 - 8^3 (ii) (−8)3+83(-8)^3 + 8^3 (iii) −83+(−8)3-8^3 + (-8)^3 (iv) 83−(−8)38^3 - (-8)^3
    5 marks
    Answer and solution

    83=5128^3 = 512 and (−8)3=−512(-8)^3 = -512

    (i) (−8)3−83=−512−512(-8)^3 - 8^3 = -512 - 512, =−1024= -1024

    (ii) (−8)3+83=−512+512(-8)^3 + 8^3 = -512 + 512, =0= 0

    (iii) −83+(−8)3=−512+(−512)-8^3 + (-8)^3 = -512 + (-512), =−512−512= -512 - 512, =−1024= -1024

    (iv) 83−(−8)3=512−(−512)8^3 - (-8)^3 = 512 - (-512), =512+512= 512 + 512, =1024= 1024

    Answer: (i) −1024-1024 (ii) 00 (iii) −1024-1024 (iv) 10241024