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

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A community garden in Perth is designing a new square vegetable plot using identical rectangular raised beds. Each raised bed is 66 centimetres long and 91 centimetres wide.

The plot is to be formed by arranging 66 beds along one side and 91 beds along the adjacent side, creating a large rectangular area. The total area of the plot is calculated as (66 × 91) cm × (66 × 91) cm.

Using exponent laws, determine the exact factor by which the total area of the plot is greater than the area of a single raised bed. 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 area of a single raised bed is calculated as length × width: 66 cm × 91 cm.
  2. Substitute values: 66 × 91 = 6006 cm². So, the area of one bed is 6006 cm².
  3. The total plot is formed by arranging 66 beds along one side and 91 beds along the adjacent side, so the total number of beds is 66 × 91.
  4. The total area of the plot is (66 × 91) cm × (66 × 91) cm, which is (66 × 91)² cm².
  5. The factor by which the total area exceeds the area of one bed is: total area ÷ area of one bed.
  6. Substitute: (66 × 91)² ÷ (66 × 91) = (66 × 91)¹ = 66 × 91.
  7. Apply exponent law: a^m ÷ a^n = a^(m−n). Here, (66×91)² ÷ (66×91)¹ = (66×91)^(2−1) = (66×91)¹.
  8. So, the factor is 66 × 91. Compute this product.
  9. Calculate 66 × 91 using distributive property: 66 × (90 + 1) = 66×90 + 66×1.
  10. 66×90 = 5940; 66×1 = 66. Add: 5940 + 66 = 6006.
  11. The factor is 6006. This is the exact factor by which the total area is greater than one bed's area.
  12. Check: (66×91)² ÷ (66×91) = 66×91 = 6006. Correct.

Reference answer: 6006

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