NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Exponent laws

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A craft designer in Brisbane is creating a symmetrical mosaic using square tiles. Each tile has a side length of 28 millimetres.

The design requires a large square formation where each side of the square is made up of 68 tiles placed edge to edge. Using exponent laws, determine the exact ratio of the area of the large square formation to the area of a single tile.

Express your answer as a single integer.

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Join repeated factors. x³ × x² has three x-factors followed by two more. That is five factors of x, so x³ × x² = x⁵.xxxxx3 factors2 factorsx³ × x² = x⁵
x³ × x² has three x-factors followed by two more. That is five factors of x, so x³ × x² = x⁵.

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Worked solution

  1. The side length of one tile is 28 mm. The area of a single tile is calculated as side length squared: 28 mm × 28 mm.
  2. Calculate the area of one tile: 28^2 = 784 mm².
  3. The large square formation has 68 tiles along one side. The total number of tiles in the formation is 68 × 68.
  4. Calculate the total number of tiles: 68^2 = 4624 tiles.
  5. The total area of the large square formation is the number of tiles multiplied by the area of one tile: 4624 × 784 mm².
  6. Express the total area using exponent laws: (68^2) × (28^2) = (68 × 28)^2.
  7. The ratio of the large square area to one tile area is: (68 × 28)^2 / (28^2).
  8. Apply the quotient of powers rule: (68 × 28)^2 / 28^2 = 68^2 × 28^2 / 28^2.
  9. Cancel 28^2 in numerator and denominator: 68^2 × 1 = 68^2.
  10. Calculate 68^2: 68 × 68 = 4624.
  11. The ratio of the area of the large square to one tile is 4624, which is an integer.
  12. Check: 68^2 = 4624, and since the area scales with the square of the side length, and the side length is 68 times larger, the area ratio is 68^2 = 4624. This confirms the result.

Reference answer: 4624

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