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Question 1130807: If (w^4 v^5)/(u^3) is greater than 0, which of the following does not have to be positive?
A. u^9 v^7 w^2
B. w^6 v^9 u^6
C. v^6 w^8 u^12
D. w^6 v^11 u^19
E. w^10 v^2 u^2
Answer by greenestamps(13209)   (Show Source):
You can put this solution on YOUR website!
We only need to count the number of negative factors in a whole expression (numerator and denominator) to determine whether the expression value is positive or negative. An even number of negative factors makes the expression value positive; an odd number makes the expression value negative.
The total number of factors in the given expression is 12; and it is given that the value of that expression is positive.
In every one of the given answer choices, there is an even number of factors of w; so the product of the factors of w is positive in every answer choice. That means the sign of the expression is always determined by the number of factors of u and v.
The sum of the numbers of factors of u and v in the given expression is 8, an even number. That means any other expression in which the sum of the factors of u and v is even will have the same sign as the given expression.
The sum of the numbers of factors of u and v in all the answer choices except B is even, so all of those answer choices must be positive.
ANSWER: The only answer choice that does not have to be positive is B.
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