Is x y ? (1) x^(1/2) y^(1/2) (2) x^2 y^2

Question

Is x > y ?

(1)  \sqrt{x}>\sqrt{y} 
(2)  x^2>y^2  -

Konstantin Lynov · Answer

All we know from (1) is that X and Y are positive (i.e. it could be an integer or a fraction). Consider an example:

X=4, Y=9 => sqrt(X)<sqrt(Y);
X=1/4, Y=1/9 => sqrt(X)>sqrt(Y);

Therefore, (1) alone is clearly not sufficient.

From (2) X and Y could be either positive or negative, thus, (2) alone is also not sufficient.

Combine => X&Y are positive. 

Looks like the answer is E.

AkamaiBrah · Answer

IMO, I think you need to re=examine your analysis for condition (1).

stolyar · Answer

You should start from data sets to answer the question, not vice versa.

(1) square root X > square root Y

X and Y are nonnegative

sqrt 9> sqrt 4 => 3>2 => X>Y
sqrt 1/4 > sqrt 1/9 => 1/2>1/3 => X>Y
sqrt 100 > sqrt 1/9 => 10>1/3 => X>Y
sqrt 1/4 > sqrt 0 => 1/2>0 => X>Y

More than enough

(2) clearly not sufficient because of even powers

9>4 => 3>2 => X>Y OR
9>4 => -3<2 => X<Y

Finally, A.

Sourabh2012 · Answer

Explaination by stolyar is perfect.

Sqrt of X is greater than X if X < 1 but here the comparison is between 2 different numbers X and Y. So A is sufficient.

Bunuel · Answer

USEFUL TO KNOW A. We can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality). For example: 2<4 --> we can square both sides and write: 2^2<4^2 ; 0\leq{x}<{y} --> we can square both sides and write: x^2-2 if we square we'll get 1>4 which is not right. So if given that x>y then we can not square both sides and write x^2>y^2 if we are not certain that both x and y are non-negative. B. We can always raise both parts of an inequality to an odd power (the same for taking an odd root of both sides of an inequality). For example: -2<-1 --> we can raise both sides to third power and write: -2^3=-8<-1=-1^3 or -5<1 --> -5^2=-125<1=1^3 ; x we can raise both sides to third power and write: x^3y? (1) \sqrt{x}>\sqrt{y} --> as both parts of the inequality are non-negative then according to A we can square them --> x>y . Sufficient. (2) x^2>y^2 --> |x|>|y| --> x is farther from zero than y , but this info is insufficient to say whether x>y (if x=2 and y=1 - YES but x=-2 and y=1 - NO). Not sufficient. Answer: A.

kanusha · Answer

pls correct my approach in solving this problem..

1, square root X > square root Y
Consider value: x=9, y= 4 so, suare root of 9 = 3, AND 4 = 2, AND 9>4 so AD...
2, X^2>Y^2
here x= 1,y= 2 so 1 is not greater than 2, so

only A....

can we consider negative values in square root??

Bunuel · Answer

You cannot get sufficiency based only on one set of numbers. Thus theoretically testing only one set of numbers can give you an incorrect answer.

As for your other questions: the even roots from negative numbers are not defined for the GMAT, which means that for  \sqrt{x}  to be defined, x must be more than or equal to zero.

Hope it's clear.

nehabhasin · Answer

But what about taking negative root of a number?

So in this case, what if we take negative root of X and positive root of Y? In that case option A won't be sufficient...

Where am I going wrong?

Thanks in advance!
NB

ENGRTOMBA2018 · Answer

I am assuming that you are talking about taking the negative square of a number? Something similar to , if given x^2 > y^2 ---> \sqrt{x^2} > \sqrt{y^2} ? Note that for GMAT, square root of any positive number 'x' = \sqrt {x} \geq 0. There are no 'negative' square roots for GMAT. 2 points to note (copied from math-number-theory-88376.html): • \sqrt{x^2}=|x| , when x≤0, then \sqrt{x^2}=−x and when x≥0, then \sqrt{x^2}=x • When the GMAT provides the square root sign for an even root, such as \sqrt{x} or \sqrt {x} , then the only accepted answer is the positive root. That is, \sqrt{25}=5 , NOT +5 or -5. In contrast, the equation x^2=25 has TWO solutions, +5 and -5. Even roots have only a positive value on the GMAT. Hope this helps.

nehabhasin · Answer

But what about taking negative root of a number?

So in this case, what if we take negative root of X and positive root of Y? In that case option A won't be sufficient...

Where am I going wrong?

Thanks in advance!
NB[/quote]

I am assuming that you are talking about taking the negative square of a number? Something similar to , if given x^2 > y^2 ---> \(\sqrt{x^2} > \sqrt{y^2}\) ?

Note that for GMAT, square root of any positive number 'x' = \(\sqrt {x}\) \(\geq\) 0. There are no 'negative' square roots for GMAT.

2 points to note (copied from math-number-theory-88376.html):

• \(\sqrt{x^2}=|x|\), when x≤0, then \(\sqrt{x^2}=−x\) and when x≥0, then \(\sqrt{x^2}=x\)

• When the GMAT provides the square root sign for an even root, such as \(\sqrt{x}\) or \(\sqrt[4]{x}\), then the only accepted answer is the positive root.

That is, \(\sqrt{25}=5\), NOT +5 or -5. In contrast, the equation \(x^2=25\) has TWO solutions, +5 and -5. Even roots have only a positive value on the GMAT.

Hope this helps.[/quote]

Got it! Thanks!

LogicGuru1 · Answer

Is x>y? (1) \sqrt{x}>\sqrt{y} The output of a square root is always a single positive root ; ONE UNIQUE POSITIVE ROOT VALUE \sqrt{X} >\sqrt{Y} ; therefore X>Y SUFFICIENT (2) x^2>y^2 For all practical purposes any number n^2 can be seen as | n | and will always have a positive and negative root : TWO UNIQUE ROOT VALUES (+VE , -VE) Since X can be bigger or smaller than Y (depending upon the relative sign of x and y); Nothing can be said with confidence example -5^2 > 4^2 but -5 < 4 example -3^2 < -4^2 but -3> -4 INSUFFICIENT ANSWER IS ONLY A

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

Is x > y ? (1) x^(1/2) > y^(1/2) (2) x^2 > y^2

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards