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if-a-b-c-and-d-are-integers-and-ab2c3d4-0-which-of-the-136450.htmlIf a, b, c, and d are integers and \(ab^2c^3d^4 > 0\), which of the following must be positive?
I. \(a^2cd\)
II. \(bc^4d\)
III. \(a^3c^3d^2\)A) I only
B) II only
C) III only
D) I and III
E) I, II, and III
Since given that \(a*b^2*c^3*d^4 > 0\), then we know that none of the unknowns is zero. Therefore, \(b^2>0\) and \(d^4>0\), which means that we can safely reduce by them to get \(a*c^3>0\) (so, the given expression does not depend on the value of \(b\) or \(d\): they can be positive as well as negative).
Next, \(a*c^3>0\) means that \(a\) and \(c\) must have the same sign: they are either both positive or both negative.
Evaluate each option:
I. \(a^2cd\). Since \(d\) can positive as well as negative then this option is not necessarily positive.
II. \(bc^4d\). Since \(d\) can positive as well as negative then this option is not necessarily positive.
III. \(a^3c^3d^2\). Since \(a*c^3>0\), then \(a^3*c^3>0\) and as \(d^2>0\), then their product, \((a^3*c^3)*d^2\) must be positive too.
Answer: C.