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

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Required skills (click to reveal)

A craft designer is creating a symmetrical wall art piece using square tiles. Each tile has a side length of 32 millimetres.

The design consists of a large square formation where each side of the large square is made up of 56 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 and answer (click to reveal)

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Worked solution

  1. The side length of a single tile is 32 mm. The area of one tile is calculated as side length squared: 32^2 mm².
  2. Calculate 32^2: 32 × 32 = 1024 mm². So, the area of one tile is 1024 mm².
  3. The large square formation has 56 tiles along one side. The total side length of the large square is 56 × 32 mm.
  4. Calculate the side length of the large square: 56 × 32 = 1792 mm.
  5. The area of the large square is side length squared: 1792^2 mm².
  6. Express both areas using exponent notation. Area of large square = 1792^2, area of one tile = 32^2.
  7. The ratio of the areas is (1792^2) / (32^2). This simplifies to (1792 / 32)^2 by the quotient of powers rule: (a^m / b^m) = (a/b)^m.
  8. Calculate 1792 ÷ 32. 32 × 56 = 1792, so 1792 / 32 = 56.
  9. Now compute 56^2 using exponent laws: 56 × 56 = 3136.
  10. The ratio of the area of the large square to one tile is 3136.
  11. Check: 32^2 = 1024, 1792^2 = (32 × 56)^2 = 32^2 × 56^2 = 1024 × 3136 = 3,211,264 mm². Then 3,211,264 / 1024 = 3136. Correct.
  12. The exact ratio is 3136, as required.

Reference answer: 3136

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