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Cubes and Cube Roots

Question paper, Set 2

  • Set 2 of 10
  • 20 questions
  • 50 marks
  • 1 hour 30 minutes

Download the PDF (with key sheet and solutions, 75.3 KB)

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

  1. Write y×y×yy \times y \times y in power form.
    1 mark
    Answer and solution

    y×y×y=y3y \times y \times y = y^3

    Read it as 'y cubed'.

    Answer: y3y^3

  2. Write 2×2×2×7×7×72 \times 2 \times 2 \times 7 \times 7 \times 7 using cubes.
    2 marks
    Answer and solution

    2×2×2=232 \times 2 \times 2 = 2^3

    7×7×7=737 \times 7 \times 7 = 7^3

    ∴2×2×2×7×7×7=23×73\therefore 2 \times 2 \times 2 \times 7 \times 7 \times 7 = 2^3 \times 7^3

    Answer: 23×732^3 \times 7^3

  3. The cube of which number is 125?
    1 mark
    Answer and solution

    5×5×5=1255 \times 5 \times 5 = 125

    ∴125=53\therefore 125 = 5^3

    Answer: 55

  4. Is 100 the cube of a whole number? Use the table of cubes.
    2 marks
    Answer and solution

    From the table: 43=644^3 = 64 and 53=1255^3 = 125.

    64<100<12564 < 100 < 125, and no whole number lies between 4 and 5.

    So 100 is not a cube.

    Answer: No, it lies between 43=644^3 = 64 and 53=1255^3 = 125

  5. A big cube is made of small cubes, with 6 small cubes along each edge. (i) How many small cubes are in one layer? (ii) How many small cubes are there in all?
    3 marks
    Answer and solution

    (i) One layer has 6 rows of 6: 6×6=366 \times 6 = 36

    (ii) There are 6 layers: 36×6=21636 \times 6 = 216

    So there are 63=2166^3 = 216 small cubes.

    Answer: (i) 3636 (ii) 216216

  6. Find 20320^3.
    2 marks
    Answer and solution

    20×20=40020 \times 20 = 400

    400×20=8000400 \times 20 = 8000

    ∴203=8000\therefore 20^3 = 8000

    Answer: 80008000

  7. Three cube-shaped stacks are built from small wooden cubes. They have 11, 13 and 20 small cubes along each edge. How many small cubes are in each stack?
    5 marks
    Answer and solution

    Edge 11: 11×11=12111 \times 11 = 121, 121×11=1331121 \times 11 = 1331

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

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

    Answer: 13311331, 21972197, 80008000 small cubes

  8. Write (−a)3(-a)^3 in terms of a3a^3.
    1 mark
    Answer and solution

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

    =−a3= -a^3

    Answer: −a3-a^3

  9. Show that (−8)3=−83(-8)^3 = -8^3.
    2 marks
    Answer and solution

    LHS =(−8)3=(−8)×(−8)×(−8)=64×(−8)=−512= (-8)^3 = (-8) \times (-8) \times (-8) = 64 \times (-8) = -512

    RHS =−83=−(8×8×8)=−512= -8^3 = -(8 \times 8 \times 8) = -512

    LHS = RHS, so (−8)3=−83(-8)^3 = -8^3.

    Answer: Both sides are −512-512

  10. Find 737^3 and (−7)3(-7)^3. How are the two answers related?
    3 marks
    Answer and solution

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

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

    The two answers are the same number with opposite signs: (−7)3=−73(-7)^3 = -7^3.

    Answer: 73=3437^3 = 343, (−7)3=−343(-7)^3 = -343; same number, opposite signs

  11. Find 43+434^3 + 4^3.
    1 mark
    Answer and solution

    43+43=64+644^3 + 4^3 = 64 + 64

    =128= 128

    Answer: 128128

  12. Find the sum of the cubes of all the even numbers from 2 to 8.
    5 marks
    Answer and solution

    23+43+63+83=8+64+216+5122^3 + 4^3 + 6^3 + 8^3 = 8 + 64 + 216 + 512

    =800= 800

    Answer: 800800

  13. Simplify m3+(−m)3m^3 + (-m)^3.
    1 mark
    Answer and solution

    (−m)3=−m3(-m)^3 = -m^3

    ∴m3+(−m)3=m3−m3=0\therefore m^3 + (-m)^3 = m^3 - m^3 = 0

    Answer: 00

  14. Find (−2)3+(−5)3(-2)^3 + (-5)^3.
    2 marks
    Answer and solution

    (−2)3+(−5)3=−8+(−125)(-2)^3 + (-5)^3 = -8 + (-125)

    =−8−125= -8 - 125

    =−133= -133

    Answer: −133-133

  15. Ravi found 43−73=2794^3 - 7^3 = 279. Is he right? Find the correct answer.
    2 marks
    Answer and solution

    43−73=64−3434^3 - 7^3 = 64 - 343

    We take away the bigger number 343 from the smaller number 64, so the answer is negative.

    64−343=−27964 - 343 = -279

    Ravi subtracted the other way round; the correct answer is −279-279.

    Answer: No. 43−73=−2794^3 - 7^3 = -279

  16. Meena built a big cube using 7 small cubes along each edge. She then took it apart and used some of the small cubes to build a smaller cube with 4 small cubes along each edge. How many small cubes were left over?
    3 marks
    Answer and solution

    At first: 73=3437^3 = 343

    At the end: 43=644^3 = 64

    Difference = 343−64=279343 - 64 = 279

    Answer: 279279 small cubes

  17. Find 103−63−8310^3 - 6^3 - 8^3.
    5 marks
    Answer and solution

    103−63−83=1000−216−51210^3 - 6^3 - 8^3 = 1000 - 216 - 512

    =272= 272

    Answer: 272272

  18. Simplify x3−(−x)3x^3 - (-x)^3.
    1 mark
    Answer and solution

    (−x)3=−x3(-x)^3 = -x^3

    x3−(−x3)=x3+x3=2x3x^3 - (-x^3) = x^3 + x^3 = 2x^3

    Answer: 2x32x^3

  19. Find −33−73-3^3 - 7^3.
    3 marks
    Answer and solution

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

    =−370= -370

    Answer: −370-370

  20. Find the value of (−3)3+53−(−2)3−43(-3)^3 + 5^3 - (-2)^3 - 4^3.
    5 marks
    Answer and solution

    (−3)3+53−(−2)3−43=−27+125−(−8)−64(-3)^3 + 5^3 - (-2)^3 - 4^3 = -27 + 125 - (-8) - 64

    =−27+125+8−64= -27 + 125 + 8 - 64

    =42= 42

    Answer: 4242