The prompt was designed by Helen Hindle (see box, right below) for year 7 mixed attainment classes. It can be shown to be true by pairing up the factors of 24:
(1 x 24) x (2 x 12) x (3 x 8) x (4 x 6) = 24 x 24 x 24 x 24 = 244
Indeed, there is a 'rule' for all numbers: the exponent is half the number of factors. So, for example, the product of the six factors of 12 equals 123. Often students take some convincing that the exponent in the case of prime numbers - that is, one - is legitimate. Furthermore, students regularly believe that there is one exception to the rule. Square numbers, they claim, do not "follow the rule because one factor is on its own." However, it turns out that square numbers do behave in the same way. For example, the product of the three factors of nine equals 91.5 and the product of the nine factors of 36 equals 364.5.
The realisation that square numbers follow the rule can form the basis for an inquiry into square roots and the representation of a square root as a fractional exponent. The inquiry can be extended further into other fractional and even negative exponents, although at this stage the link with factors is severed.
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