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

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A craft designer is creating a symmetrical wall art piece using square tiles. Each tile has a side length of 48 millimetres.

The design consists of a large square formation where each side is made up of 49 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 a single tile is 48 millimetres. The area of one tile is calculated as side length squared: 48 mm × 48 mm.
  2. Calculate the area of one tile: 48^2 = (48 × 48) = 2304 square millimetres.
  3. The large square formation has 49 tiles along one side. The total side length of the large square is 49 × 48 mm.
  4. Calculate the side length of the large square: 49 × 48 = 2352 millimetres.
  5. The area of the large square is side length squared: 2352^2 square millimetres.
  6. To find the ratio of the large square area to one tile area, divide the large square area by the tile area: 2352^2 / 48^2.
  7. Apply the exponent law for quotients: (a/b)^n = a^n / b^n. Here, (2352 / 48)^2 = (49)^2 because 2352 ÷ 48 = 49.
  8. Simplify: 49^2 = 49 × 49 = 2401.
  9. The ratio of the large square area to one tile area is 2401.
  10. Verify using exponent laws: (49 × 48)^2 / 48^2 = 49^2 × 48^2 / 48^2 = 49^2 = 2401.
  11. The result is an integer, and the calculation confirms that the area ratio is exactly 2401.
  12. The ratio of the large square area to a single tile area is 2401, which is a whole number and matches the expected form.

Reference answer: 2401

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