If a^8b^7c^6d^5 0, which of the following products must be positive

Question

If  a^8b^7c^6d^5>0 , which of the following products must be positive ?

I. ab

II. bc

III. bd

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

chetan2u · Answer

a^8b^7c^6d^5>0

a^8(b^6b)c^6(d^4d)>0

a^8b^6c^6d^4(bd)>0

Now even powers means positive and we can say anything about the sign of the base as even power will always make them positive. 
Thus, we are left with bd>0

Only III

C

Crytiocanalyst · Answer

Even powers will always produce positive numbers even if the term to be squared is negative

therefore b and d can be negative and result in a positve term when multiplied

THerefore b*d can be positive others can be argued both ways therefore out

Therefore IMO C

GMATWhizTeam · Answer

Solution:

We are given:  a^8b^7c^6d^5>0

From this we can infer that

1.  b  and  d  are positive numbers because they have odd powers over them i.e., 7 and 5 respectively, and yet there product is positive or  >0 .

2.  a  and  c  on the other hand can be positive or negative. They have even powers over them. We cannot say.

Now upon looking at the available option we see that only III i.e.,  bd  must always be positive because we know  b,d>0 .

Hence the right answer is    Option C   .

chetan2u · Answer

The coloured portion is wrong. 
 b^7d^5>0  does not mean that b and d are positive numbers. We can only say that both b and d have same sign.

HWPO · Answer

I solved it without calculating anything at all:

We know that a and c are greater than 0 no matter what - because their exponents are even.

So, it means that b and d have to help us maintain the inequality:

if b is <0 and d is >0 than we will have: positive*negative*positive*positive = negative (<0)
if b is >0 and d is <0 then we will have: positive*positive*positive*negative = negative (<0)

SO, b*d has to be positive in order to maintain the inequality: positive*positive*positive*positive = positive (>0)

UUBA · Answer

a^8 and c^6 are positive (even power), but the value of 'a' and 'c' can be either '+ve' or '-ve'. On the other side, the signs of 'b' and 'd' have to be the same for the entire product to be '+ve'. Hence bd must be positive.

If a^8b^7c^6d^5 > 0, which of the following products must be positive

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