Note: This model, however, does not explain very well powers with negative integer or rational exponents (see non-instance 4).
In the case of the area of a rectangle, the formula is A = l x w, where the length and width are not the same (as they are for a square). In this case we do not add the exponents of l and w. This concept can also be applied to the volume of a rectangular prism, where the length, width and height are not the same.
The model involving the square and the rectangle can create a better understanding of the concept that the bases of numbers with exponents need to be the same in order to apply the exponent rule where the exponents are added. There can often be a misconception that any numbers with exponents can apply to this rule and this can be further disproved through more examples, i.e.:
3⁽²⁺³⁾ = 3⁵ = 243
Note: Neither a base of 2 nor 3 with an exponent of 5 is equal to 72, so the exponent rule does not apply to different bases
This can also be further used to demonstrate the formula for the area and volume of a rectangle/rectangular prism, which also has different bases and therefore is an exception to this particular exponent law.
Instance 3a: Different bases; Bases with a LCF (lowest common factor)
2³ x 4² = 2³ x (2 x 2)2 = 2³ x (22 x 22)
= (2 x 2 x 2) x (2 x 2 x 2 x 2) = 2(3+4) = 128
Instance 3b: Different bases; Bases with a LCF (lowest common factor) and a coefficient
2³ x 6² = 2³ x (3 x 2)2 = 2³ x (32 x 22)
= (2 x 2 x 2) x (3 x 3 x 2 x 2) = 2(3+2) x 9 = 32 x 9 = 288