Find possible dimensions. Read the description below

Question

A box is shown below. To find the volume of the box, multiply the length, the width, and the height. An expression for the volume of the box is stated below. Determine possible dimensions for the length, width, and height or the box. 

Explain your thinking. Justify your work. Show ALL necessary work.

Volume Expression: 27x^10y^12

Answer 1

V=wlh = 27x^10(y^12)

w=3x^3(y^4)

l = 3x^3(y^4)

h =3x^4(y^4)

that's making the box as close to a cube, with integer exponents

the exponents of x have to sum to 10, exponents of y sum to 12

coefficients' product = 27

the coefficients could be 3, 3 and 3, or 9, 3 and 1, or 27, 1 and 1

x terms could be x, x and x^8, x^2, x^2 and x^6, or x5, x5 and x^0 (1)

y terms could be y3, y3 and y^6,

such as

w=27x^10(y^12)

l =1

h -1

etc.