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Math Expert V
Joined: 02 Sep 2009
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12 00:00

Difficulty:   85% (hard)

Question Stats: 34% (01:16) correct 66% (01:06) wrong based on 93 sessions

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Is $$a^7*b^2*c^3 \gt 0$$?

(1) $$bc \lt 0$$

(2) $$ac \gt 0$$

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Math Expert V
Joined: 02 Sep 2009
Posts: 59265

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Official Solution:

Inequality $$a^7*b^2*c^3 \gt 0$$ to be true $$a$$ and $$c$$ must be either both positive or both negative AND $$b$$ must not be zero (in order $$a^7*b^2*c^3$$ not to equal zero).

(1) $$bc \lt 0$$. Hence, $$b \ne 0$$. Don't know about $$a$$ and $$c$$. Not sufficient.

(2) $$ac \gt 0$$. Hence, $$a$$ and $$c$$ are either both positive or both negative. Don't know about $$b$$: if $$b=0$$, then the expression will be equal 0. Not sufficient.

(1)+(2) Sufficient.

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anubhavmax wrote:
Bunuel - Could you please look into why (2) alone is not sufficient ?

We have ac > 0.

$$a^7*b^2*c^3 = a^4 * b^2 * a^3 * c^3 = a^4 * b^2 * (ac)^3$$. All 3 are positive, so the whole expression is positive. B should be the answer I feel.

Hi anubhav...

How about of b=0 then the expression a^7*b^2*c*3=0
So you need information that terms are not zero..B tells you a and c have same sign but not whether b =0 or not

hope it helps
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Math Expert V
Joined: 02 Sep 2009
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mvictor wrote:
bc<0
ac>0

here we have:
either b is negative, then a and c are positive, in this case a^7*b^2*c*3<0
or b is positive, and a and c are negative, in this case a^7*b^2*c*3>0
from where did we get b not 0?

Can b be 0 if bc < 0? NO! If b = 0, then bc = 0, not < 0.
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Bunuel - Could you please look into why (2) alone is not sufficient ?

We have ac > 0.

$$a^7*b^2*c^3 = a^4 * b^2 * a^3 * c^3 = a^4 * b^2 * (ac)^3$$. All 3 are positive, so the whole expression is positive. B should be the answer I feel.
Manager  Joined: 01 Feb 2013
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Oh right... thanks a lot. Simply overlooked the zero case. Board of Directors P
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bc<0
ac>0

here we have:
either b is negative, then a and c are positive, in this case a^7*b^2*c*3<0
or b is positive, and a and c are negative, in this case a^7*b^2*c*3>0
from where did we get b not 0?
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Joined: 07 Mar 2011
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I think this is a high-quality question and I agree with explanation. I need one point clarification from Bunuel is:

when you say bc < 0 it is possible that c as well not equal to zero along with b. So we have some information on C also right.
May be I am not getting your point.
Math Expert V
Joined: 02 Sep 2009
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dharan wrote:
I think this is a high-quality question and I agree with explanation. I need one point clarification from Bunuel is:

when you say bc < 0 it is possible that c as well not equal to zero along with b. So we have some information on C also right.
May be I am not getting your point.

Yes, bc < 0 means that none of them is 0. But we need to know the sign of c not just that it's not 0, that's why we say that there is not sufficient info on c there.
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I think this is a high-quality question and I agree with explanation.
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I think this is a high-quality question and I agree with explanation.
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Bunuel wrote:
Is $$a^7*b^2*c^3 \gt 0$$?

(1) $$bc \lt 0$$

(2) $$ac \gt 0$$

Hi Bunuel

If a^7 = that can indicate A can be (+\-) going ahead , * B2 = this means B will be every time positive, C^3 , C may be +\- values

coming back to the main equation , if we multiply Negative * positive * Negative = this gives us a positive value . Is my mistake only that I haven't considered Zero in my values ? As by even / odd concept , I had marked this D .
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Math Expert V
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loserunderachiever wrote:
Bunuel wrote:
Is $$a^7*b^2*c^3 \gt 0$$?

(1) $$bc \lt 0$$

(2) $$ac \gt 0$$

Hi Bunuel

If a^7 = that can indicate A can be (+\-) going ahead , * B2 = this means B will be every time positive, C^3 , C may be +\- values

coming back to the main equation , if we multiply Negative * positive * Negative = this gives us a positive value . Is my mistake only that I haven't considered Zero in my values ? As by even / odd concept , I had marked this D .

x^odd can be positive, negative or 0.
x^even can only be positive, or 0.

Not sure I understand what's your question though.
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Bunuel wrote:
loserunderachiever wrote:
Bunuel wrote:
Is $$a^7*b^2*c^3 \gt 0$$?

(1) $$bc \lt 0$$

(2) $$ac \gt 0$$

Hi Bunuel

If a^7 = that can indicate A can be (+\-) going ahead , * B2 = this means B will be every time positive, C^3 , C may be +\- values

coming back to the main equation , if we multiply Negative * positive * Negative = this gives us a positive value . Is my mistake only that I haven't considered Zero in my values ? As by even / odd concept , I had marked this D .

x^odd can be positive, negative or 0.
x^even can only be positive, or 0.

Not sure I understand what's your question though.

I got your point , as I didn't consider X to be Zero .

Thanks.
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Bunuel wrote:
Official Solution:

Inequality $$a^7*b^2*c^3 \gt 0$$ to be true $$a$$ and $$c$$ must be either both positive or both negative AND $$b$$ must not be zero (in order $$a^7*b^2*c^3$$ not to equal zero).

(1) $$bc \lt 0$$. Hence, $$b \ne 0$$. Don't know about $$a$$ and $$c$$. Not sufficient.

(2) $$ac \gt 0$$. Hence, $$a$$ and $$c$$ are either both positive or both negative. Don't know about $$b$$: if $$b=0$$, then the expression will be equal 0. Not sufficient.

(1)+(2) Sufficient.

Hi,
How can we say 1 + 2 is sufficient?(What if a=0 or c=0)
Because we have not considered a=0 or c=0 case.
Why have we not considered this and only considered b=0 case?
Math Expert V
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Akshit03 wrote:
Bunuel wrote:
Official Solution:

Inequality $$a^7*b^2*c^3 \gt 0$$ to be true $$a$$ and $$c$$ must be either both positive or both negative AND $$b$$ must not be zero (in order $$a^7*b^2*c^3$$ not to equal zero).

(1) $$bc \lt 0$$. Hence, $$b \ne 0$$. Don't know about $$a$$ and $$c$$. Not sufficient.

(2) $$ac \gt 0$$. Hence, $$a$$ and $$c$$ are either both positive or both negative. Don't know about $$b$$: if $$b=0$$, then the expression will be equal 0. Not sufficient.

(1)+(2) Sufficient.

Hi,
How can we say 1 + 2 is sufficient?(What if a=0 or c=0)
Because we have not considered a=0 or c=0 case.
Why have we not considered this and only considered b=0 case?

How can either of them be 0 if it's given that ac > 0?
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Intern  B
Joined: 26 Nov 2018
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I think this is a high-quality question. I solved it like this:
a^7*b^2*c^3=a^6*(bc)^2* ac
using both statements; bc<0;so, (bc)^2 >0 & ac>0

also a^6 being even power will be positive so the value as a whole will be greater than zero. Re M11-26   [#permalink] 22 Apr 2019, 20:44
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# M11-26

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