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Multiple choice physics dynamics - explaining motion exponentiation index notation and products of prime factors powers

A calorie is a unit of heat energy and its value is 4.18 J where $1 J = 1 kg m^2 s^{-2}$. Suppose we use a new system of units in which unit of mass equals $\alpha$ kg, the unit of length equals $\beta$ m and the unit of the time is $\gamma$ sec. Then the value of a calorie in the new system of units is then

  1. 4.18 $\displaystyle \frac{\gamma^2}{\alpha \beta^2}$
  2. 4.18 $\displaystyle \frac{\alpha \beta^2}{\gamma^2}$
  3. 4.18 $\displaystyle \frac{\gamma^2}{\alpha}$
  4. 4.18 $\displaystyle \frac{\beta^2}{\alpha \gamma^2}$
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A Correct answer
Explanation

$1 J = (1 kg) ( 1 m)^2 (1 sec)^{-2}$
$1 x = (\alpha kg) (\beta m)^2 (\gamma sec)^{-2}$
$\displaystyle \therefore \frac{1 J}{1x} = \left( \frac{1}{\alpha} \right) \left( \frac{1}{\beta} \right)^2 (\gamma)^2 = \frac{\gamma^2}{\alpha \beta^2} $
$\displaystyle \therefore 1 J = \frac{\gamma^2}{\alpha \beta^2}$ or $\displaystyle 1 cal = 4.18 \displaystyle \frac{\gamma^2}{\alpha \beta^2}$