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

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

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11 Apr 2012, 14:52
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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.
[Reveal] Spoiler: OA
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Joined: 02 Sep 2009
Posts: 43867
Re: If r, s, and t are nonzero integers, is r^5*s^3*t^4 negative [#permalink]

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11 Apr 2012, 15:11
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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.

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Hope it helps.
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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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13 Jun 2012, 10:56
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Hi,

$$r^5s^3t^4 < 0$$?

Using (1);
rt is -ve,
so, $$r^5s^3t^4 = (rt)^4r^1s^3$$, no clue about s, Insufficient.

Using (2),
s is -ve, r & t are unknown, Insufficient.

Using (1) & (2),
$$(rt)^4r^1s^3$$
$$(rt)^4 > 0$$, and $$s < 0$$
so the expression reduces to$$r^1(-ve)$$, but we still don't know if r is +ve/-ve.

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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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05 Sep 2012, 04:25
Once you realize that two exponents are odd and one is even, you know that the you only have to worry about the signs of r and s, thus reducing the complexity.
Picking numbers might also help, I think, even though it might take a bit too long.

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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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31 Jan 2018, 03:59
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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Joined: 02 Sep 2009
Posts: 43867
Re: If r, s, and t are nonzero integers, is r^5*s^3*t^4 negative [#permalink]

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31 Jan 2018, 04:08
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]

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31 Jan 2018, 11:36
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.

The given expression can be written as -
$$(rt)^4*r*s^3$$

Statement I:

$$rt = -ve$$.... In $$(rt)^4*r*s^3$$, we don't know anything about r & s... So, Insufficient.

Statement II:

$$s = -ve$$. But still we dont know anything about $$r$$.

Combined, still we don't know about $$r$$ sign.

Hence, E.
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Re: If r, s, and t are nonzero integers, is r^5*s^3*t^4 negative   [#permalink] 31 Jan 2018, 11:36
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