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(1) From this, x = y^2, so by substitution then x^y = (y^2)^y or y^(2y), and y^x = y^(y^2). Comparing x^y to y^x can then be done by comparing y^(2y) to y^(y^2), or simply comparing the exponents 2y to y^2. If, for example, y = 2, then 2y = 4 and y^2 = 4, and then x^y would equal y^x. If, however, y = 3, then 2y = 6 and y^2 = 9, and so x^y would be less than y^x; NOT SUFFICIENT.
(2) It is known that y > 2, but no information about x is given; NOT SUFFICIENT.
If both (1) and (2) are taken together, then 2y is compared to y^2 (1) and from (2) it is known that y > 2, so 2y will always be less than y^2. Therefore, x^y < y^x.
Re: Exponents/inequalities problem from QR 2nd DS 121 [#permalink]
04 Aug 2010, 05:21
This is a tricky question. I think it relies on you misapplying the rule: \((x^a)^b = x^{ab}\). Is this only valid if a and b are constants?
Example: (1) \(x = y^2\);
\(x^y < y^x\) therefore, \((y^2)^y < y^{y^2}\). How do you simplify this? The guide shows to \(y^{2y} < y^{y^2}\). The left hand side makes sense to me.
Why would \(y^{y^2}\) not simplify to \(y^{2y}\) also? Plugging in numbers, it makes sense. I just want to understand the concept.
Re: Exponents/inequalities problem from QR 2nd DS 121 [#permalink]
04 Aug 2010, 05:52
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jpr200012 wrote:
If x and y are nonzero integers, is \(x^y < y^x\)?
(1) \(x = y^2\) (2) \(y > 2\)
If x and y are nonzero integers, is \(x^y < y^x\)?
(1) \(x = y^2\) --> if \(x=y=1\), then \(x^y=1=y^x\), so the answer would be NO BUT if \(y=3\) and \(x=9\), then \(x^y=9^3<y^x=3^9\), so the answer would be YES. Not sufficient.
(2) \(y>2\). No info about \(x\), not sufficient.
(1)+(2) From (1) \(x = y^2\), thus the question becomes: is \((y^2)^y<y^{(y^2)}\)? --> is \(y^{2y}<y^{(y^2)}\)? Now, since from (2) \(y=integer>2\), then \(2y\) will always be less than \(y^2\), therefore \(y^{2y}\) will be less than \(y^{(y^2)}\). Sufficient.
Answer: C.
jpr200012 wrote:
This is a tricky question. I think it relies on you misapplying the rule: \((x^a)^b = x^{ab}\). Is this only valid if a and b are constants?
Example: (1) \(x = y^2\);
\(x^y < y^x\) therefore, \((y^2)^y < y^{y^2}\). How do you simplify this? The guide shows to \(y^{2y} < y^{y^2}\). The left hand side makes sense to me.
Why would \(y^{y^2}\) not simplify to \(y^{2y}\) also? Plugging in numbers, it makes sense. I just want to understand the concept.
If exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus: \(a^m^n=a^{(m^n)}\) and not \((a^m)^n\), which on the other hand equals to \(a^{mn}\).
Re: Exponents/inequalities problem from QR 2nd DS 121 [#permalink]
05 Aug 2010, 01:25
Posting this msg here even though i sent a private msg to you-for the benefit of others here.
Hi Bunuel, apprecite ur wonderful explanation. I am having trouble in DS question where x & y are termed as non-zero integers.
What is the best way to analyze instances where x & y are are NEGATIVE integers. I see that u have not analyzed this possibility. is there a trick to be sure that this is not needed as u have solved in this case?
Re: Exponents/inequalities problem from QR 2nd DS 121 [#permalink]
05 Aug 2010, 02:12
1
This post received KUDOS
Expert's post
ramanankris wrote:
Posting this msg here even though i sent a private msg to you-for the benefit of others here.
Hi Bunuel, apprecite ur wonderful explanation. I am having trouble in DS question where x & y are termed as non-zero integers.
What is the best way to analyze instances where x & y are are NEGATIVE integers. I see that u have not analyzed this possibility. is there a trick to be sure that this is not needed as u have solved in this case?
Please enlighten. Thanks.
On DS questions when plugging numbers, goal is to prove that the statement is not sufficient. So we should try to get a YES answer with one chosen number(s) and a NO with another.
For statement (1) I got YES answer and then NO answer with positive numbers, so my goal to prove that this statement was not sufficient was reached, hence there was no need to try negative numbers.
x=y^2 tells us that x is positive, but it tells us nothing about y.
For example if y=1 then x=1. Therefore 1^1=1^1 and x^y=y^x If y=-2 then x=4. Therefore 4^-2<-2^4 Since we cannot get a definite yes or no from this statement, it is INSUFFICIENT
Statement 2: Insufficient y>2 This tells us nothing about x. If x=-1 and y=4, then -1^4>4^-1 If x=5 and y=3, then 5^3<3^5 Since we cannot get a definite yes or no from this statement, it is INSUFFICIENT
Putting both statements together we know that y>2 and x=y^2 If y=4, then x=16, then 16^4<4^16 (16^4 = 4^8). No matter which integers you choose x^y < y^x, so Statements 1 and 2 together are SUFFICIENT. The answer is C. _________________
The Brain Dump - From Low GPA to Top MBA(Updated September 1, 2013) - A Few of My Favorite Things--> http://cheetarah1980.blogspot.com
if x and y are nonzero integers is, x^y < y^x? (1) x = y^2 (2) y > 2
x^y < y^x Stmt1: x=y^2 \(x^y ---> y^2^y= y^(2y)\) \(y^x ---> y^(y^2)\) Is y^(2y) < y^(y^2) ? Take log both side 2y logy < y^2logy ? Canceling log y Is 2y < y^2 ? Is 2<y ? i.e Is y>2? We don't know the value of y. Hence not sufficient.
Stmt2: y>2 Not sufficient.
Combining, from Stmt2: we know that y>2 . Hence Stmt1: Is y>2 can be answered taking Stmt1 and Stmt2 together.
OA C. _________________
My dad once said to me: Son, nothing succeeds like success.
if x and y are nonzero integers is, x^y < y^x? (1) x = y^2 (2) y > 2
x^y < y^x Stmt1: x=y^2 \(x^y ---> y^2^y= y^(2y)\) \(y^x ---> y^(y^2)\) Is y^(2y) < y^(y^2) ? Take log both side 2y logy < y^2logy ? Canceling log y Is 2y < y^2 ? Is 2<y ? i.e Is y>2? We don't know the value of y. Hence not sufficient.
Stmt2: y>2 Not sufficient.
Combining, from Stmt2: we know that y>2 . Hence Stmt1: Is y>2 can be answered taking Stmt1 and Stmt2 together.
OA C.
Nice solve. However, GMAT Math does not require the knowledge of logarithms. Definitely can help on the test, but for people who haven't touched a logarithm since high school it's not necessary to relearn them to answer this question. _________________
The Brain Dump - From Low GPA to Top MBA(Updated September 1, 2013) - A Few of My Favorite Things--> http://cheetarah1980.blogspot.com
************************ If y>0; y-2>0; Is y>2 OR y<0 ***********************
However, "y<0" actually doesn't hold true for x^y<y^x (for y=-1) *********************
The only point I am trying to make here is that solving through logarithm may give us undesired results. What if x^y or y^x is negative. Then, taking the logarithms would be wrong!!!
On another look, From stmt1: \(x=y^2\), so\(y=\sqrt{x}\). Since \(\sqrt{x}\)will always be a positive number, y will always be positive. Substituting, in \(x^y < y^x\), \(x^sqrt(x)\) < \(\sqrt{x} ^ x\) Now we can safely take log as both sides are positive. \(sqrt(x)logx < xlog sqrt(x)\) \(sqrt(x) logx < x/2 log x\) Is \(sqrt(x) < x/2\) ? Is \(2 < sqrt(x)\) ? Cannot be determined. Not sufficient.
Stmt2: y>2. Not sufficient.
Combining, from stmt1 \(y=\sqrt{x}\) From stmt2: y>2. i.e \(\sqrt{x}\) >2. Hence \(2 < sqrt(x)\) ? . Yes.
OA C. _________________
My dad once said to me: Son, nothing succeeds like success.
On another look, From stmt1: \(x=y^2\), so\(y=\sqrt{x}\). Since \(\sqrt{x}\)will always be a positive number, y will always be positive. Substituting, in \(x^y < y^x\), \(x^sqrt(x)\) < \(\sqrt{x} ^ x\) Now we can safely take log as both sides are positive. \(sqrt(x)logx < xlog sqrt(x)\) \(sqrt(x) logx < x/2 log x\) Is \(sqrt(x) < x/2\) ? Is \(2 < sqrt(x)\) ? Cannot be determined. Not sufficient.
Stmt2: y>2. Not sufficient.
Combining, from stmt1 \(y=\sqrt{x}\) From stmt2: y>2. i.e \(\sqrt{x}\) >2. Hence \(2 < sqrt(x)\) ? . Yes.
OA C.
x=y^2, Just because x is positive doesn't mean y has to be positive. Even powers always yield a positive number, even if the base is negative. For example, if x=4 then y can be 2 or -2. The only thing we can determine from statement 1 is that x is positive. y can be either positive or negative. If y is negative then x^y would be 1/x^y. 1/x^y may or may not be greater than y^x. y=-3, x=9 vs y=3, x=9. 1/9^3>-3^9. 9^3<3^9. _________________
The Brain Dump - From Low GPA to Top MBA(Updated September 1, 2013) - A Few of My Favorite Things--> http://cheetarah1980.blogspot.com
Re: If x and y are nonzero integers, is x^y < y^x? (1) x = [#permalink]
20 Jul 2014, 10:19
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