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# If r, s, and t are nonzero integers, is r^5*s^3*t^4 negative

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Re: If r, s, and t are nonzero integers, is r^5*s^3*t^4 negative [#permalink]
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$$r^5s^3t^4$$ can be re-written as $$(rt)^2(rst)^2*rs$$

Now $$(rt)^2$$ and $$(rst)^2$$ are always positive. The question boils down to IS rs negative?

Stmt1: rt is negative. Either r is -ve or t is -ve. No info about s. INSUFF
Stmt2: No info about r. INSUFF
Together: still no info about r, it could be +ve or -ve. E is the solution.
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Re: If r, s, and t are nonzero integers, is r^5*s^3*t^4 negative [#permalink]
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Bunuel wrote:
dzodzo85 wrote:
If r, s, and t are nonzero integers, is r^5*s^3*t^4 negative?

(1) rt is negative
(2) s is negative

Could someone kindly render a more easily understood explanation than that found in the Quantitative Review 2nd Edition. It would be really appreciated.

Since r, s, and t are nonzero integers then in order r^5*s^3*t^4 to be negative, only one condition should hold: r and s must have the opposite signs, in this case (r^5*s^3)*t^4=(negative)*(positive)=negative. Notice that if we were not told that given variables are nonzero then there would be one more condition that t must not be zero.

(1) rt is negative --> r and t have the opposite signs. Not sufficient, since no info about s.
(2) s is negative. Clearly insufficient.

(1)+(2) If r is positive then the answer will be YES (since r^5*s^3*t^4=positive*negative*positive=negative) but if r is negative then the answer will be NO (r^5*s^3*t^4=negative*negative*positive=positive). Not sufficient.

Hope it helps.

For (1)+(2), how could R be positive? Isn't T automatically positive because it is T^4?

Statement 1 says RT is negative, so I assumed that R has to be negative. I'm sure this is an elementary question, but can you please show me an example of how T^4 could be negative?

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Re: If r, s, and t are nonzero integers, is r^5*s^3*t^4 negative [#permalink]
msurls wrote:
Bunuel wrote:
dzodzo85 wrote:
If r, s, and t are nonzero integers, is r^5*s^3*t^4 negative?

(1) rt is negative
(2) s is negative

Could someone kindly render a more easily understood explanation than that found in the Quantitative Review 2nd Edition. It would be really appreciated.

Since r, s, and t are nonzero integers then in order r^5*s^3*t^4 to be negative, only one condition should hold: r and s must have the opposite signs, in this case (r^5*s^3)*t^4=(negative)*(positive)=negative. Notice that if we were not told that given variables are nonzero then there would be one more condition that t must not be zero.

(1) rt is negative --> r and t have the opposite signs. Not sufficient, since no info about s.
(2) s is negative. Clearly insufficient.

(1)+(2) If r is positive then the answer will be YES (since r^5*s^3*t^4=positive*negative*positive=negative) but if r is negative then the answer will be NO (r^5*s^3*t^4=negative*negative*positive=positive). Not sufficient.

Hope it helps.

For (1)+(2), how could R be positive? Isn't T automatically positive because it is T^4?

Statement 1 says RT is negative, so I assumed that R has to be negative. I'm sure this is an elementary question, but can you please show me an example of how T^4 could be negative?

t^4 cannot be negative. A number in an even power is always non-negative, so 0 or positive. Since we are told that t is nonzero, then t^4 is positive only. But t itself could be positive as well as negative. For example, t^4 = 16 = positive, t = 2 or t = -2.
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Re: If r, s, and t are nonzero integers, is r^5*s^3*t^4 negative [#permalink]
HiBunuel
I understood the solution but could you expand on this a little more?
Quote:
Notice that if we were not told that given variables are nonzero then there would be one more condition that t must not be zero.

Just for improving my insight.

Thank you,
Dablu
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Re: If r, s, and t are nonzero integers, is r^5*s^3*t^4 negative [#permalink]
gurudabl wrote:
HiBunuel
I understood the solution but could you expand on this a little more?
Quote:
Notice that if we were not told that given variables are nonzero then there would be one more condition that t must not be zero.

Just for improving my insight.

Thank you,
Dablu

If we were not told that given variables are nonzero, then r and s having the opposite signs would not be enough for r^5*s^3*t^4 to be negative. In that case, we should also check whether t is not 0, because even if r and s will have the opposite signs but t = 0, then r^5*s^3*t^4 =0, so it's not negative.

Hope it's clear.
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Re: If r, s, and t are nonzero integers, is r^5*s^3*t^4 negative [#permalink]
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Re: If r, s, and t are nonzero integers, is r^5*s^3*t^4 negative [#permalink]
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