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

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A sports equipment supplier in Melbourne is packing cricket balls into storage units. Each box contains 82 cricket balls, and there are 46 identical boxes.

The storage units are designed to hold 82^2 cricket balls each. Using exponent laws, determine the exact number of storage units required to hold all the cricket balls.

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 total number of cricket balls is calculated by multiplying the number of balls per box by the number of boxes: 82 × 46.
  2. Perform the multiplication: 82 × 46 = 3772. This is the total number of cricket balls.
  3. Each storage unit holds 82^2 cricket balls. Calculate 82^2 using exponent notation: 82 × 82.
  4. Compute 82 × 82: 80×80 = 6400, 80×2 = 160, 2×80 = 160, 2×2 = 4; sum: 6400 + 160 + 160 + 4 = 6724. So, 82^2 = 6724.
  5. The number of storage units required is the total balls divided by balls per unit: 3772 ÷ 6724.
  6. Simplify the fraction 3772 / 6724 using exponent laws. Note that 3772 = 82 × 46 and 6724 = 82^2.
  7. Apply the quotient of powers rule: 82^1 / 82^2 = 82^(1−2) = 82^(-1). But since we are dividing 82 × 46 by 82^2, rewrite as (82 × 46) / (82 × 82) = 46 / 82.
  8. Simplify 46 / 82 by dividing numerator and denominator by their greatest common divisor, 2: 46 ÷ 2 = 23, 82 ÷ 2 = 41. So, 46 / 82 = 23 / 41.
  9. Since 23 and 41 are both prime and have no common factors, the fraction 23/41 is in simplest form.
  10. The result 23/41 is not an integer, but the question asks for the number of storage units required. Since storage units must be whole, and 23/41 < 1, only one full unit is needed to hold all balls, but it will not be full.
  11. However, the question specifies to use exponent laws to determine the exact number of storage units. Since 3772 / 6724 = 23/41, and 23/41 < 1, the number of full storage units required is 0, but this contradicts the need to hold all balls.
  12. Re-evaluate: the exponent law approach shows that 82 × 46 / 82^2 = 46 / 82 = 23 / 41. But since 23/41 < 1, and we cannot have a fraction of a storage unit, we must use ceiling division. However, the question says 'determine the exact number' and 'express as a single integer'. Since 23/41 is not an integer, and the division does not yield a whole number, the only way to hold all balls is to use 1 storage unit. But 1 × 6724 = 6724 > 3772, so 1 unit is sufficient. The exact number of storage units required is 1.

Reference answer: 1

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