NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Exponent laws

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A cube has a side length of 4 centimetres. Another cube has a side length of 12 centimetres.

Using exponent laws, determine the exact factor by which the volume of the larger cube is greater than the volume of the smaller cube. Express your answer as a single integer.

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Explanation

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Worked solution

  1. The volume of a cube is given by V = s^3, where s is the side length in centimetres.
  2. The smaller cube has side length 4 cm, so its volume is 4^3 = 4 × 4 × 4.
  3. Calculate 4^3: 4 × 4 = 16, then 16 × 4 = 64. So volume of smaller cube is 64 cm³.
  4. The larger cube has side length 12 cm, so its volume is 12^3 = 12 × 12 × 12.
  5. Calculate 12^3: 12 × 12 = 144, then 144 × 12 = 1728. So volume of larger cube is 1728 cm³.
  6. To find the factor by which the volume increases, divide the larger volume by the smaller: 1728 ÷ 64.
  7. Use exponent laws: 12^3 / 4^3 = (12/4)^3 = 3^3.
  8. Simplify 12/4 = 3, so (12/4)^3 = 3^3.
  9. Calculate 3^3: 3 × 3 = 9, then 9 × 3 = 27.
  10. The volume of the larger cube is 27 times greater than that of the smaller cube.
  11. Check: 64 × 27 = 1728. Correct, since 64 × 20 = 1280, 64 × 7 = 448, 1280 + 448 = 1728.
  12. The factor is exactly 27, as required.

Reference answer: 27

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