math hw help pls?

Question

If the product of the integers a, b, and c is 1, then what is the difference between the largest and smallest possible values of a^2 x b^3 x c^4?

a) -2 b) -1 c) 0 d)1 e)2
pls explain or show work

Answer 1

If a*b*c = 1, and they're all integers, then either
(1) all three of a,b,c must equal 1, or else
(2) two of the three must equal -1 and the third must equal 1.

If any of a, b, or c were zero, the product would be zero. If any were greater than 1 or less than -1, they'd have to be multiplied by a non-integer to bring the product to 1. (For example if a=2, then bc must equal 1/2 for the product to be one, but you can't find a product of two integers that is 1/2.)

Since a^2 * b^3 * c^4 has an even power for a and c, it doesn't matter whether a or c is -1 or 1, they contribute 1 to the product. That is to say, the product equals 1 * b^3 * 1, which is just b^3.

If b=1, then b^3 = 1, and the product is 1.
If b=-1, then b^3=-1, and the product is -1.

Thus, the largest possible product (1) minus the smallest (-1) is 2, which is your answer (e).