If a, b, c, and d are integers and ab2c3d4 > 0, which of the

I think we can realize real quick that this question is about identifying what variable might be negative and what has to be positive. We know that the given variables with their exponents will result in a positive. i.e. larger than 0. we know "a" has to be positive since we don't know if "b" might be negative or not. C is positive because the exponent is an odd exponent, and we don't know if "d" is a positive or negative. With this information, lets go down the answer choice.

Hmm, there is an issue with option 1. Because we don't know if "d" is a positive or negative, c*d could be a negative. so thats out.

Same principle with option 2. We don't know if "d" is negative or positive, so thats out too.

Which leaves us with only option 3, but lets take a quick look. We know "a" and "c" is positive, so the odd exponent is no issue, and since "d" has an even exponent, we know for sure that it must be positive.

a*(b^2)*(c^3)*(d^4) > 0,

since b^2 and d^4 are always positive..

a*(c^3) is positive .. so 'a' and 'c' are of the same sign, eithr positive or negative. we have no information about 'b' and 'd' , so they could b negative..

so the first 2 statements ( I and II ) have 'd' in them, which could be negative.. so we can eliminate them then and there.

in statement III, 'd' is squared, so thats positive and since 'a' and 'c' are of the same sign, a^3 * c^3 must be positive.

so III only , that option C.

If a, b, c, and d are integers and ab2c3d4>0, which of the following must be positive?

Because b2d4 will always be positive, ac3 is positive which means, a & c are either both positive or negative

I. a2cd => Can not say because we don't know about d
II. bc4d=> Can not say because we don't know about b & d
III. a3c3d2=> It must be positive because d2 will always be positive & on behalf of the given condition we know that ac is also positive

Therefore, Only III is true

Hence C