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Cubes and Cube Roots

Question paper, Set 4

  • Set 4 of 10
  • 20 questions
  • 50 marks
  • 1 hour 30 minutes

Download the PDF (with key sheet and solutions, 74.8 KB)

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

  1. Write a3a^3 as a product.
    1 mark
    Answer and solution

    a3a^3 means aa multiplied by itself 3 times.

    ∴a3=a×a×a\therefore a^3 = a \times a \times a

    Answer: a×a×aa \times a \times a

  2. Is 535^3 the same as 5×35 \times 3? Find both values.
    2 marks
    Answer and solution

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

    5×3=155 \times 3 = 15

    535^3 means 5 multiplied by itself 3 times, not 5 times 3.

    Answer: No. 53=1255^3 = 125, but 5×3=155 \times 3 = 15

  3. Find the cube of 4.
    1 mark
    Answer and solution

    43=4×4×44^3 = 4 \times 4 \times 4

    =16×4=64= 16 \times 4 = 64

    Answer: 6464

  4. A big cube is built from small cubes, with 5 small cubes along each edge. How many small cubes are used in all?
    2 marks
    Answer and solution

    Number = 5×5×55 \times 5 \times 5

    =53=25×5=125= 5^3 = 25 \times 5 = 125

    Answer: 125125 small cubes

  5. Fill in the boxes using the table of cubes: (i) □3=512\square^3 = 512 (ii) 93=□9^3 = \square (iii) □3=125\square^3 = 125
    3 marks
    Answer and solution

    (i) 8×8×8=5128 \times 8 \times 8 = 512, so the box is 8.

    (ii) 93=9×9×9=7299^3 = 9 \times 9 \times 9 = 729

    (iii) 5×5×5=1255 \times 5 \times 5 = 125, so the box is 5.

    Answer: (i) 88 (ii) 729729 (iii) 55

  6. Find the cube of 15.
    2 marks
    Answer and solution

    15×15=22515 \times 15 = 225

    153=225×15=337515^3 = 225 \times 15 = 3375

    Answer: 33753375

  7. Find 232^3, 20320^3 and 2003200^3. How many zeros does 20320^3 have more than 232^3, and 2003200^3 more than 20320^3?
    5 marks
    Answer and solution

    23=2×2×2=82^3 = 2 \times 2 \times 2 = 8

    203=20×20×20=400×20=800020^3 = 20 \times 20 \times 20 = 400 \times 20 = 8000

    2003=200×200×200=40000×200=8000000200^3 = 200 \times 200 \times 200 = 40000 \times 200 = 8000000

    Each time the number is multiplied by 10, its cube gets 3 more zeros.

    Answer: 88, 80008000, 80000008000000; 3 more zeros each time

  8. Without working it out, say whether (−13)3(-13)^3 is positive or negative. Give the reason.
    1 mark
    Answer and solution

    (−13)3=−133(-13)^3 = -13^3

    The cube of a negative number is negative.

    Answer: Negative

  9. Find (−6)3(-6)^3 and (−10)3(-10)^3.
    2 marks
    Answer and solution

    (−6)3=−63=−216(-6)^3 = -6^3 = -216

    10×10=10010 \times 10 = 100, 100×10=1000100 \times 10 = 1000

    ∴(−10)3=−1000\therefore (-10)^3 = -1000

    Answer: (−6)3=−216(-6)^3 = -216, (−10)3=−1000(-10)^3 = -1000

  10. Find the cubes of −7-7, −9-9 and −20-20.
    3 marks
    Answer and solution

    (−7)3=−73=−(49×7)=−343(-7)^3 = -7^3 = -(49 \times 7) = -343

    (−9)3=−93=−(81×9)=−729(-9)^3 = -9^3 = -(81 \times 9) = -729

    (−20)3=−203=−(400×20)=−8000(-20)^3 = -20^3 = -(400 \times 20) = -8000

    Answer: (−7)3=−343(-7)^3 = -343, (−9)3=−729(-9)^3 = -729, (−20)3=−8000(-20)^3 = -8000

  11. Add the cube of 6 to itself.
    1 mark
    Answer and solution

    63+63=216+2166^3 + 6^3 = 216 + 216

    =432= 432

    Answer: 432432

  12. Three big cubes are built from small cubes, with 3, 4 and 6 small cubes along each edge. How many small cubes are used in all?
    5 marks
    Answer and solution

    Small cubes used = 33+43+633^3 + 4^3 + 6^3

    33+43+63=27+64+2163^3 + 4^3 + 6^3 = 27 + 64 + 216

    =307= 307

    Answer: 307307 small cubes

  13. Find the value of (−8)3+83(-8)^3 + 8^3.
    1 mark
    Answer and solution

    (−8)3=−83(-8)^3 = -8^3

    (−8)3+83=83−83=0(-8)^3 + 8^3 = 8^3 - 8^3 = 0

    Answer: 00

  14. Find the value of x3+(−y)3x^3 + (-y)^3 when x=6x = 6 and y=2y = 2.
    2 marks
    Answer and solution

    x3+(−y)3=63+(−2)3x^3 + (-y)^3 = 6^3 + (-2)^3

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

    =216−8= 216 - 8

    =208= 208

    Answer: 208208

  15. What must be added to 434^3 to get 636^3?
    2 marks
    Answer and solution

    Number to be added = 63−436^3 - 4^3

    63−43=216−646^3 - 4^3 = 216 - 64

    =152= 152

    Answer: 152152

  16. What must be subtracted from 11311^3 to get 838^3?
    3 marks
    Answer and solution

    Number = 113−8311^3 - 8^3

    113=121×11=133111^3 = 121 \times 11 = 1331

    113−83=1331−51211^3 - 8^3 = 1331 - 512

    =819= 819

    Answer: 819819

  17. How much more is 939^3 than 53+635^3 + 6^3?
    5 marks
    Answer and solution

    93=7299^3 = 729

    53+63=125+216=3415^3 + 6^3 = 125 + 216 = 341

    Difference = 729−341=388729 - 341 = 388

    Answer: 388388

  18. Find (−1)3×63(-1)^3 \times 6^3.
    1 mark
    Answer and solution

    (−1)3=−1(-1)^3 = -1, 63=2166^3 = 216

    −1×216=−216-1 \times 216 = -216

    Answer: −216-216

  19. If x=−4x = -4 and y=3y = 3, find x3−y3x^3 - y^3.
    3 marks
    Answer and solution

    x3−y3=(−4)3−33x^3 - y^3 = (-4)^3 - 3^3

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

    =−91= -91

    Answer: −91-91

  20. If a=4a = 4, b=−5b = -5 and c=3c = 3, find a3+b3+c3a^3 + b^3 + c^3.
    5 marks
    Answer and solution

    a3+b3+c3=43+(−5)3+33a^3 + b^3 + c^3 = 4^3 + (-5)^3 + 3^3

    43+(−5)3+33=64+(−125)+274^3 + (-5)^3 + 3^3 = 64 + (-125) + 27

    =64−125+27= 64 - 125 + 27

    =−34= -34

    Answer: −34-34