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

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A community garden in Melbourne is divided into rectangular plots. One plot has a length of 43 metres and a width of 27 metres.

The gardeners plan to install a pathway that runs along the diagonal of this plot. Using the exponent laws to simplify the expression for the diagonal length derived from the Pythagorean theorem, calculate the exact length of the diagonal in metres.

The expression for the diagonal is sqrt(43^2 + 27^2). Simplify this expression using exponent laws and basic arithmetic, and give the exact result as a single simplified radical expression.

Rectangle measurementsLength 43 metres; width 27 metres43 metres27 metres
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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.
Four squared. 4² = 4 × 4 = 16 unit squares.4 units across4
4² = 4 × 4 = 16 unit squares.

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Worked solution

  1. The diagonal of a rectangle can be found using the Pythagorean theorem: d = sqrt(l^2 + w^2), where l is the length and w is the width.
  2. Substitute the given values: l = 43 m, w = 27 m. So, d = sqrt(43^2 + 27^2).
  3. Calculate 43^2: 43 × 43 = 1849.
  4. Calculate 27^2: 27 × 27 = 729.
  5. Add the squares: 1849 + 729 = 2578.
  6. Now, d = sqrt(2578). This is the exact expression for the diagonal length.
  7. Factor 2578 to check for perfect square factors. Begin by testing divisibility by small primes.
  8. 2578 is even, so divide by 2: 2578 ÷ 2 = 1289.
  9. Check if 1289 is divisible by any square number. 1289 is not divisible by 4, 9, 25, 49, 121, or 169. It is not a perfect square.
  10. Since 1289 is a prime number (verified by testing divisibility up to sqrt(1289) ≈ 35.9), and 2 is not a square factor in the remaining product, no perfect square factors exist in 2578 other than 1.
  11. Therefore, sqrt(2578) cannot be simplified further. The expression is already in simplest radical form.
  12. The exact length of the diagonal is sqrt(2578) metres.

Reference answer: sqrt(2578)

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