Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 50000

If the sum of the square roots of two integers is
[#permalink]
Show Tags
30 Oct 2016, 07:36
Question Stats:
68% (02:28) correct 32% (02:45) wrong based on 645 sessions
HideShow timer Statistics




Director
Joined: 05 Mar 2015
Posts: 995

Re: If the sum of the square roots of two integers is
[#permalink]
Show Tags
30 Oct 2016, 10:00
Bunuel wrote: If the sum of the square roots of two integers is \(\sqrt{9+6\sqrt{2}}\), what is the sum of the squares of these two integers?
(A) 40 (B) 43 (C) 45 (D) 48 (C) 52 Let nos be x &y √x + √y= \(\sqrt{9+6\sqrt{2}}\) sq both sides. x+y+2√xy=9+6√2 since x & y are integers x+y=9(1) and 2√xy=6√2 or √xy=3√2 sq both sides to get xy=18(2) sq . both sides (1) x^2+y^2+2xy=81 x^2+y^2=812xy x^2+y^2=8136=45(as xy=18 from (2)) Ans C




Manager
Status: Quant Expert Q51
Joined: 02 Aug 2014
Posts: 65

Re: If the sum of the square roots of two integers is
[#permalink]
Show Tags
30 Oct 2016, 12:18
Let a and b be both of the integers.\(\sqrt{a}+\sqrt{b}=\sqrt{9+6\sqrt{2}}\) Lets square both sides of the equation we get \(a+b+2\sqrt{a}\sqrt{b}=9+6\sqrt{2}\) Then \(a+b= 9\) [1] \(2\sqrt{a}\sqrt{b}=6\sqrt{2}\) [2] [2] \(\sqrt{ab}=3\sqrt{2}\) lets square both sides \(ab=18\) so we get a system \(a+b=9\) \(ab=18\) Combining both equations we get : \(a^29a+18=0\) Solving this second degree equation we get : \(a = 3\) and \(b = 6\) We are searching for the sum of the squares of these two integers.
so \(a^2+b^2=9+36 = 45\) So the answer is C.
_________________
Cours particuliers de GMAT



Intern
Joined: 30 Jun 2017
Posts: 7
Location: India

If the sum of the square roots of two integers is
[#permalink]
Show Tags
17 Jul 2017, 11:57
AnisMURR wrote: Let a and b be both of the integers.
\(\sqrt{a}+\sqrt{b}=\sqrt{9+6\sqrt{2}}\)
Lets square both sides of the equation
we get
\(a+b+2\sqrt{a}\sqrt{b}=9+6\sqrt{2}\)
Then
\(a+b= 9\) [1]
\(2\sqrt{a}\sqrt{b}=6\sqrt{2}\) [2]
[2] \(\sqrt{ab}=3\sqrt{2}\) lets square both sides \(ab=18\)
so we get a system
\(a+b=9\) \(ab=18\)
Combining both equations we get : \(a^29a+18=0\)
Solving this second degree equation we get : \(a = 3\) and \(b = 6\)
We are searching for the sum of the squares of these two integers.
so \(a^2+b^2=9+36 = 45\)
So the answer is C. I don't think this method will be helpful in GMAT  where we target a problem not more than 2 min. Just try this one.. we know that sqaure of integers can only be from terms of the series of 1,4,9,16,25,36,49,64,....... Further, summation of any two terms from the series should be equal to the one of the options given. It comes out that only 40 (36+4) and 45 (36+9) can be formed from the series of square of integers. By ballparking sqaure root of complex number given comes out to be square root of 18 i.e. slightly more than 4. whereas the summation of sqaure root of 2 & 6 is slightly less than 4 and the summation of sqaure root of 3 & 6 is slightly more than 4. Hence answer is C.



Manager
Status: Quant Expert Q51
Joined: 02 Aug 2014
Posts: 65

Re: If the sum of the square roots of two integers is
[#permalink]
Show Tags
19 Jul 2017, 23:02
Hello Metwing Nice analysis But beleive me it took me less than 2 minutes. Best,
_________________
Cours particuliers de GMAT



Intern
Joined: 25 Apr 2017
Posts: 15

Re: If the sum of the square roots of two integers is
[#permalink]
Show Tags
02 Aug 2017, 16:17
AnisMURR wrote: \(a+b=9\) \(ab=18\)
Combining both equations we get : \(a^29a+18=0\)
Please how do you arrive at the above equation from those 2? Can't seem to figure it out. Seems like a step is missing  as a expert, it is probably obvious to you. But after 30 minutes, I am still clueless.



Senior Manager
Joined: 24 Apr 2016
Posts: 333

Re: If the sum of the square roots of two integers is
[#permalink]
Show Tags
02 Aug 2017, 16:36
getitdoneright wrote: AnisMURR wrote: \(a+b=9\) \(ab=18\)
Combining both equations we get : \(a^29a+18=0\)
Please how do you arrive at the above equation from those 2? Can't seem to figure it out. Seems like a step is missing  as a expert, it is probably obvious to you. But after 30 minutes, I am still clueless. a+b = 9 square both sides \((a+b)^2 = 9^2\) \(a^2 + b^2 + 2ab = 81\) Substituting the value of ab (18) in the above equation \(a^2 + b^2 + (2*18) = 81\) \(a^2 + b^2 = 81  36 = 45\) Hope this helps



Manager
Joined: 07 Jun 2017
Posts: 103

Re: If the sum of the square roots of two integers is
[#permalink]
Show Tags
02 Aug 2017, 22:00
rohit8865 wrote: Bunuel wrote: If the sum of the square roots of two integers is \(\sqrt{9+6\sqrt{2}}\), what is the sum of the squares of these two integers?
(A) 40 (B) 43 (C) 45 (D) 48 (C) 52 Let nos be x &y √x + √y= \(\sqrt{9+6\sqrt{2}}\) sq both sides. x+y+2√xy=9+6√2 since x & y are integers x+y=9(1) and 2√xy=6√2 or √xy=3√2 sq both sides to get xy=18(2) sq . both sides (1) x^2+y^2+2xy=81 x^2+y^2=812xy x^2+y^2=8136=45(as xy=18 from (2)) Ans C Dear, How do you get "x^2+y^2+2xy=81"? Where is this 81 from? Thank you so much.



Manager
Joined: 19 Aug 2016
Posts: 86

If the sum of the square roots of two integers is
[#permalink]
Show Tags
05 Aug 2017, 17:54
pclawong wrote: rohit8865 wrote: Bunuel wrote: If the sum of the square roots of two integers is \(\sqrt{9+6\sqrt{2}}\), what is the sum of the squares of these two integers?
(A) 40 (B) 43 (C) 45 (D) 48 (C) 52 Let nos be x &y √x + √y= \(\sqrt{9+6\sqrt{2}}\) sq both sides. x+y+2√xy=9+6√2 since x & y are integers x+y=9(1) and 2√xy=6√2 or √xy=3√2 sq both sides to get xy=18(2) sq . both sides (1) x^2+y^2+2xy=81 x^2+y^2=812xy x^2+y^2=8136=45(as xy=18 from (2)) Ans C Dear, How do you get "x^2+y^2+2xy=81"? Where is this 81 from? Thank you so much. In the above equation, we have got x+y=9 (eqn 1)so when u square on both sides u will get x^2+y^2+2xy=81



Senior Manager
Status: love the club...
Joined: 24 Mar 2015
Posts: 267

Re: If the sum of the square roots of two integers is
[#permalink]
Show Tags
30 Sep 2017, 20:08
Bunuel wrote: If the sum of the square roots of two integers is \(\sqrt{9+6\sqrt{2}}\), what is the sum of the squares of these two integers?
(A) 40 (B) 43 (C) 45 (D) 48 (C) 52 hi Bunuel very high quality question this one is indeed. Can you please provide some links to such questions to practice..? thanks in advance, man



Math Expert
Joined: 02 Sep 2009
Posts: 50000

Re: If the sum of the square roots of two integers is
[#permalink]
Show Tags
01 Oct 2017, 03:59



Senior Manager
Status: love the club...
Joined: 24 Mar 2015
Posts: 267

Re: If the sum of the square roots of two integers is
[#permalink]
Show Tags
01 Oct 2017, 08:50
Bunuel wrote: gmatcracker2017 wrote: Bunuel wrote: If the sum of the square roots of two integers is \(\sqrt{9+6\sqrt{2}}\), what is the sum of the squares of these two integers?
(A) 40 (B) 43 (C) 45 (D) 48 (C) 52 hi Bunuel very high quality question this one is indeed. Can you please provide some links to such questions to practice..? thanks in advance, man Roots DS QuestionsRoots PS QuestionsHope it helps. thanks man great you are



Manager
Joined: 19 Aug 2016
Posts: 152
Location: India
GPA: 3.82

Re: If the sum of the square roots of two integers is
[#permalink]
Show Tags
05 Nov 2017, 12:24
hi Bunuel very high quality question this one is indeed. Can you please provide some links to such questions to practice..? thanks in advance, man[/quote][/quote] Hello, I'm still unable to understand the solution. Could you please provide the official solution or another explaination to the question? Thanks
_________________
Consider giving me Kudos if you find my posts useful, challenging and helpful!



Senior SC Moderator
Joined: 22 May 2016
Posts: 2033

If the sum of the square roots of two integers is
[#permalink]
Show Tags
06 Nov 2017, 10:20
Bunuel wrote: If the sum of the square roots of two integers is \(\sqrt{9+6\sqrt{2}}\), what is the sum of the squares of these two integers?
(A) 40 (B) 43 (C) 45 (D) 48 (C) 52 AnisMURR wrote: Let a and b be both of the integers.
\(\sqrt{a}+\sqrt{b}=\sqrt{9+6\sqrt{2}}\)
Lets square both sides of the equation
we get
\(a+b+2\sqrt{a}\sqrt{b}=9+6\sqrt{2}\)
Then
\(a+b= 9\) [1]
\(2\sqrt{a}\sqrt{b}=6\sqrt{2}\) [2]
[2] \(\sqrt{ab}=3\sqrt{2}\) lets square both sides \(ab=18\)
so we get a system
\(a+b=9\) \(ab=18\)
Combining both equations we get : \(a^29a+18=0\)
Solving this second degree equation we get : \(a = 3\) and \(b = 6\)
We are searching for the sum of the squares of these two integers.
so \(a^2+b^2=9+36 = 45\)
So the answer is C. AnisMURR , I can follow everything if I accept this part's last line: Quote: \(\sqrt{a}+\sqrt{b}=\sqrt{9+6\sqrt{2}}\)
Lets square both sides of the equation
we get
\(a+b+2\sqrt{a}\sqrt{b}=9+6\sqrt{2}\) It looks as if you've gotten to a version of a square of a sum (?): \((a + b)^2 = a^2 + 2ab + b^2\) Why does (a + b) = 9? Put another way, why is there not a separate "b" (or analogous b^2?) term? I think I am missing something really obvious.
_________________
The only thing more dangerous than ignorance is arrogance.  Albert Einstein



Target Test Prep Representative
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2830

Re: If the sum of the square roots of two integers is
[#permalink]
Show Tags
08 Nov 2017, 17:35
Bunuel wrote: If the sum of the square roots of two integers is \(\sqrt{9+6\sqrt{2}}\), what is the sum of the squares of these two integers?
(A) 40 (B) 43 (C) 45 (D) 48 (C) 52 We can let a = the first integer and b = the second integer. Thus: √a + √b = √(9 + 6√2) We are asked to find a^2 + b^2. Let’s square both sides of the equation above. (√a + √b)^2 = [√(9 + 6√2)]^2 a + 2√ab + b = 9 + 6√2 Since a and b are integers, we must have: a + b = 9 and 2√ab = 6√2 If we square both sides of a + b = 9, we have: a^2 + 2ab + b^2 = 81 If we square both sides of 2√ab = 6√2, we have: 4ab = 36(2) 2ab = 36 We can now substitute 36 for 2ab in a^2 + 2ab + b^2 = 81 to obtain: a^2 + 36 + b^2 = 81 a^2 + b^2 = 45 Answer: C
_________________
Jeffery Miller
Head of GMAT Instruction
GMAT Quant SelfStudy Course
500+ lessons 3000+ practice problems 800+ HD solutions



Director
Joined: 27 May 2012
Posts: 583

Re: If the sum of the square roots of two integers is
[#permalink]
Show Tags
22 Feb 2018, 07:17
JeffTargetTestPrep wrote: Bunuel wrote: If the sum of the square roots of two integers is \(\sqrt{9+6\sqrt{2}}\), what is the sum of the squares of these two integers?
(A) 40 (B) 43 (C) 45 (D) 48 (C) 52 We can let a = the first integer and b = the second integer. Thus: √a + √b = √(9 + 6√2) We are asked to find a^2 + b^2. Let’s square both sides of the equation above. (√a + √b)^2 = [√(9 + 6√2)]^2 a + 2√ab + b = 9 + 6√2 Since a and b are integers, we must have: a + b = 9 and 2√ab = 6√2 If we square both sides of a + b = 9, we have: a^2 + 2ab + b^2 = 81 If we square both sides of 2√ab = 6√2, we have: 4ab = 36(2) 2ab = 36 We can now substitute 36 for 2ab in a^2 + 2ab + b^2 = 81 to obtain: a^2 + 36 + b^2 = 81 a^2 + b^2 = 45 = Answer: C Just curious if the individual values of the two integers are 6 and 3 or 3 and 6 respectively then of course on squaring both \(6^2 + 3^2 = 45\) but how on taking square root of 6 and 3 and summing them do we get \(\sqrt {9 + 6\sqrt{2}}\)
_________________
 Stne



PS Forum Moderator
Joined: 25 Feb 2013
Posts: 1216
Location: India
GPA: 3.82

Re: If the sum of the square roots of two integers is
[#permalink]
Show Tags
22 Feb 2018, 09:20
stne wrote: JeffTargetTestPrep wrote: Bunuel wrote: If the sum of the square roots of two integers is \(\sqrt{9+6\sqrt{2}}\), what is the sum of the squares of these two integers?
(A) 40 (B) 43 (C) 45 (D) 48 (C) 52 We can let a = the first integer and b = the second integer. Thus: √a + √b = √(9 + 6√2) We are asked to find a^2 + b^2. Let’s square both sides of the equation above. (√a + √b)^2 = [√(9 + 6√2)]^2 a + 2√ab + b = 9 + 6√2 Since a and b are integers, we must have: a + b = 9 and 2√ab = 6√2 If we square both sides of a + b = 9, we have: a^2 + 2ab + b^2 = 81 If we square both sides of 2√ab = 6√2, we have: 4ab = 36(2) 2ab = 36 We can now substitute 36 for 2ab in a^2 + 2ab + b^2 = 81 to obtain: a^2 + 36 + b^2 = 81 a^2 + b^2 = 45 = Answer: C Just curious if the individual values of the two integers are 6 and 3 or 3 and 6 respectively then of course on squaring both \(6^2 + 3^2 = 45\) but how on taking square root of 6 and 3 and summing them do we get \(\sqrt {9 + 6\sqrt{2}}\) Hi stneHere it is said that SUM of square root of integer equals \(\sqrt {9 + 6\sqrt{2}}\) Now a funny thing about the SUM is that you can arrive at a particular SUM by using various combination. For eg. if I say SUM of two integer is 9, then it can be 6+3 also or 8+1. So we have 6+3=9=8+1 but 6, 8, 3 & 1 are all different. So if \(\sqrt{6}+\sqrt{3}=4.18154055\), so is \(\sqrt {9 + 6\sqrt{2}}=4.18154055\) Hence here we will have to look at the totality and not the individual elements. I also believe that there might be a way to simplify \(\sqrt{6}+\sqrt{3}\), and get \(\sqrt {9 + 6\sqrt{2}}\)



Director
Joined: 27 May 2012
Posts: 583

Re: If the sum of the square roots of two integers is
[#permalink]
Show Tags
22 Feb 2018, 12:25
niks18 wrote: Hi stneHere it is said that SUM of square root of integer equals \(\sqrt {9 + 6\sqrt{2}}\) Now a funny thing about the SUM is that you can arrive at a particular SUM by using various combination. For eg. if I say SUM of two integer is 9, then it can be 6+3 also or 8+1. So we have 6+3=9=8+1 but 6, 8, 3 & 1 are all different. So if \(\sqrt{6}+\sqrt{3}=4.18154055\), so is \(\sqrt {9 + 6\sqrt{2}}=4.18154055\) Hence here we will have to look at the totality and not the individual elements. I also believe that there might be a way to simplify \(\sqrt{6}+\sqrt{3}\), and get \(\sqrt {9 + 6\sqrt{2}}\) Hi niks18, Really appreciate your reply,thanks a ton,maybe we have to work our way backwards to arrive at our query. Maybe some one will show us the way some day.Thanks again.
_________________
 Stne



Director
Joined: 17 Dec 2012
Posts: 629
Location: India

If the sum of the square roots of two integers is
[#permalink]
Show Tags
08 Mar 2018, 15:53
Bunuel wrote: If the sum of the square roots of two integers is \(\sqrt{9+6\sqrt{2}}\), what is the sum of the squares of these two integers?
(A) 40 (B) 43 (C) 45 (D) 48 (C) 52 Main Idea:Make the LHS correspond to RHS Details : Let the integers be x and y. We have sqrt(x) + sqrt(y) = sqrt(9+6*sqrt(2)) Squaring both sides, we have x+y+2 *sqrt(xy) =9+6*sqrt(2). 6*sqrt(2) can be written as 2*sqrt(18) So we have x+y+2 *sqrt(xy)=9+2*sqrt(18) LHs and RHS correspond . We see x+y=9 and xy=18 Solving we have x=3 and y=6 x^2 +y^2 = 36 +9 =45 Hence C.
_________________
Srinivasan Vaidyaraman Sravna Holistic Solutions http://www.sravnatestprep.com
Holistic and Systematic Approach



Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 6385
GPA: 3.82

Re: If the sum of the square roots of two integers is
[#permalink]
Show Tags
24 Apr 2018, 12:21
Bunuel wrote: If the sum of the square roots of two integers is \(\sqrt{9+6\sqrt{2}}\), what is the sum of the squares of these two integers?
(A) 40 (B) 43 (C) 45 (D) 48 (C) 52 √( 9 + 6√2) = √(9 + 2√18) = √6 + √3 6^2 + 3^2 = 36 + 9 = 45 The following property is applied. \(\sqrt{a+b+2\sqrt{ab}} = \sqrt{a} + \sqrt{b}\)
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The oneandonly World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only $99 for 3 month Online Course" "Free Resources30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons  try it yourself"




Re: If the sum of the square roots of two integers is &nbs
[#permalink]
24 Apr 2018, 12:21






