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Properties of Numbers - Exponents

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Properties of Numbers - Exponents

by SavageDetective » Sat Oct 19, 2013 4:28 am
Hi, I've come across this question on OG which I can't make sense of:

If r, s and t are nonzero integers, is r^5s^3t^4 negative?
(1) rt is negative
(2) s is negative

I got it right when working out that both alone were not sufficient. However, I assumed the statements together would be sufficient as r in statement 1 would have to be negative.

However, in the answer it is said that from the 2 statements r could be either negative or positive. However for r to be positive rt would not be negative, as a -t would be positive, therefore r must be negative. Makes sense?

Another thing, in the answer explanation it says that r^5s^3t^4 = (rt)^4rs^3. What is this rule?

All help would be much appreciated. Also, where can I get more practice for properties of numbers? So far I've only got 1 out of 5 of the answers right (PS and DS) - and the explanation on the book and my other resources don't go into the nitty gritty I need.

I'm using GMAT Prep Now + Purplemaths + OG + OG Quant.

Thanks a lot for the help!
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by Uva@90 » Sat Oct 19, 2013 5:05 am
SavageDetective wrote:Hi, I've come across this question on OG which I can't make sense of:

If r, s and t are nonzero integers, is r^5s^3t^4 negative?
(1) rt is negative
(2) s is negative

I got it right when working out that both alone were not sufficient. However, I assumed the statements together would be sufficient as r in statement 1 would have to be negative.

However, in the answer it is said that from the 2 statements r could be either negative or positive. However for r to be positive rt would not be negative, as a -t would be positive, therefore r must be negative. Makes sense?

Another thing, in the answer explanation it says that r^5s^3t^4 = (rt)^4rs^3. What is this rule?

All help would be much appreciated. Also, where can I get more practice for properties of numbers? So far I've only got 1 out of 5 of the answers right (PS and DS) - and the explanation on the book and my other resources don't go into the nitty gritty I need.

I'm using GMAT Prep Now + Purplemaths + OG + OG Quant.

Thanks a lot for the help!
Hi,
TO find: r^5*s^3*t^4 is negative or not ==>which can be written as = r^4*r*s^3*t^4
Since all the terms are in multiplication you can group to common power,while you cant do if they are in addition.
So, (r*t)^4 *r*s^3 (I hope I answered your question)

So we need value of r and s(both) to find negative or not.(r*t)^4 will always positive since it is raised to the power even number.

Coming to solution,
Statement 1: rt is negative

No info about S.

Hence Insufficient.

Statement 2: S is negative,

No info about r

Hence insufficient.

Combining 1 and 2:

s is negative and rt is negative -- It has to possibilities

One -> S is negative and (r positive and t negative)
here, our equation((r*t)^4 *r*s^3) will become (positive*positive*negative = Negative)
Two -> s negative and (r negative and t positive)
here, our equation((r*t)^4 *r*s^3) will become (positive*negative*negative = positive)

we are getting contradictory conclusion hence Answer is E

I hope it helps you.

Regards,
Uva.
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by theCodeToGMAT » Sat Oct 19, 2013 6:32 am
r,s,t are NON ZERO

To find: r^5s^3t^4

Rephrase it: rs<0?


Statement 1: rt is negative
R T
- +
+ -
INSUFFICIENT

Statement 2: S is negative
No info for "r"
INSUFFICIENT

Combining...
S = -ve
r = -ve or +ve
INSUFFICIENT

Answer [spoiler]{E}[/spoiler]
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by Brent@GMATPrepNow » Sat Oct 19, 2013 7:42 am
SavageDetective 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
Target question: Is (r^5)(s^3)(t^4) < 0?

IMPORTANT: When working with inequalities, we can safely divide both sides by a variable as long as we are certain that we are dividing by a positive value.

NOTE: if r, s and t are nonzero integers, then r^4 is ALWAYS POSITIVE, s^2 is ALWAYS POSITIVE, and t^4 is ALWAYS POSITIVE,

So, let's take the target inequality, (r^5)(s^3)(t^4) < 0, and divide both sides by r^4 to get: (r)(s^3)(t^4) < 0
Then divide both sides by s^2 to get: (r)(s)(t^4) < 0
And divide both sides by t^4 to get: (r)(s) < 0

This means we can rephrase our target question as...
REPHRASED target question: Is rs < 0?

Aside: We have a free video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

Statement 1: rt is negative
There are several values of r, s and t that satisfy this condition. Here are two:
Case a: r = 1, s = -1, t = -1, in which case rs < 0
Case b: r = -1, s = -1, t = 1, in which case rs > 0
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: s is negative
There are several values of r, s and t that satisfy this condition. Here are two:
Case a: r = 1, s = -1, t = -1, in which case rs < 0
Case b: r = -1, s = -1, t = 1, in which case rs > 0
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
There are several values of r, s and t that satisfy both statements. Here are two:
Case a: r = 1, s = -1, t = -1, in which case rs < 0
Case b: r = -1, s = -1, t = 1, in which case rs > 0
Since we cannot answer the REPHRASED target question with certainty, the combined statements are NOT SUFFICIENT

Answer = E

Cheers,
Brent
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by SavageDetective » Mon Oct 21, 2013 12:48 am
Thanks a lot guys, I'm overwhelmed with the many responses and ways to approach the problem that you guys discussed! I will go through each of them again in my next study session tonight!!

Thanks again!
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