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 Brisbane is divided into rectangular plots. One plot has a length of 95 metres and a width of 48 metres.

The gardeners plan to install a pathway 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(95^2 + 48^2). Simplify this expression using exponent laws and basic arithmetic, and give the exact result as a single simplified radical expression.

Rectangle measurementsLength 95 metres; width 48 metres95 metres48 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 = 95 m, w = 48 m. So, d = sqrt(95^2 + 48^2).
  3. Calculate 95^2: 95 × 95 = (100 - 5)^2 = 100^2 - 2×100×5 + 5^2 = 10000 - 1000 + 25 = 9025.
  4. Calculate 48^2: 48 × 48 = (50 - 2)^2 = 50^2 - 2×50×2 + 2^2 = 2500 - 200 + 4 = 2304.
  5. Add the squares: 9025 + 2304 = 11329.
  6. Now evaluate sqrt(11329). Check if 11329 is a perfect square by testing known squares near it.
  7. Note that 106^2 = 11236 and 107^2 = 11449. Since 11236 < 11329 < 11449, 11329 is not a perfect square.
  8. Factor 11329 to see if it has square factors. Check divisibility by small primes.
  9. 11329 ÷ 7 = 1618.428... not integer. ÷11 = 1029.909... not integer. ÷13 = 871.46... not integer. ÷17 = 666.41... not integer. ÷19 = 596.26... not integer. ÷23 = 492.56... not integer. ÷29 = 390.65... not integer. ÷31 = 365.45... not integer. ÷37 = 306.18... not integer. ÷41 = 276.31... not integer. ÷43 = 263.46... not integer. ÷47 = 241.04... not integer. ÷53 = 213.75... not integer. ÷59 = 191.97... not integer. ÷61 = 185.72... not integer. ÷67 = 169.08... not integer. ÷71 = 159.56... not integer. ÷73 = 155.19... not integer. ÷79 = 143.40... not integer. ÷83 = 136.49... not integer. ÷89 = 127.3... not integer. ÷97 = 116.79... not integer.
  10. Try 103^2 = 10609, 104^2 = 10816, 105^2 = 11025, 106^2 = 11236, 107^2 = 11449. 11329 - 11236 = 93, so not a square.
  11. Since 11329 has no square factors (after testing all primes up to sqrt(11329) ≈ 106.4), it cannot be simplified further. Therefore, sqrt(11329) is already in simplest radical form.
  12. The diagonal length is exactly sqrt(11329) metres. This is the simplified radical expression.

Reference answer: sqrt(11329)

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