GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 28 Jan 2020, 13:32

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# If the sum of the square roots of two integers is

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 60727
If the sum of the square roots of two integers is  [#permalink]

### Show Tags

30 Oct 2016, 07:36
12
89
00:00

Difficulty:

45% (medium)

Question Stats:

69% (02:30) correct 31% (02:45) wrong based on 713 sessions

### HideShow timer Statistics

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

_________________
Target Test Prep Representative
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2804
Re: If the sum of the square roots of two integers is  [#permalink]

### Show Tags

08 Nov 2017, 17:35
4
5
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

_________________

# Jeffrey Miller

Head of GMAT Instruction

Jeff@TargetTestPrep.com
181 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Director
Joined: 05 Mar 2015
Posts: 963
Re: If the sum of the square roots of two integers is  [#permalink]

### Show Tags

30 Oct 2016, 10:00
17
9
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=81-2xy
x^2+y^2=81-36=45---(as xy=18 from (2))

Ans C
##### General Discussion
Manager
Status: Quant Expert Q51
Joined: 02 Aug 2014
Posts: 102
Re: If the sum of the square roots of two integers is  [#permalink]

### Show Tags

30 Oct 2016, 12:18
12
5
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 : $$a^2-9a+18=0$$

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.
_________________
Intern
Joined: 30 Jun 2017
Posts: 7
Location: India
Schools: Babson '20 (A)
If the sum of the square roots of two integers is  [#permalink]

### Show Tags

17 Jul 2017, 11:57
3
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 : $$a^2-9a+18=0$$

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.
Senior Manager
Joined: 24 Apr 2016
Posts: 313
Re: If the sum of the square roots of two integers is  [#permalink]

### Show Tags

02 Aug 2017, 16:36
3
getitdoneright wrote:
AnisMURR wrote:
$$a+b=9$$
$$ab=18$$

Combining both equations we get : $$a^2-9a+18=0$$

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: 19 Aug 2016
Posts: 72
If the sum of the square roots of two integers is  [#permalink]

### Show Tags

05 Aug 2017, 17:54
1
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=81-2xy
x^2+y^2=81-36=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
Retired Moderator
Joined: 25 Feb 2013
Posts: 1156
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
1
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
=

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 stne

Here 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 $$\sqrt {9 + 6\sqrt{2}}$$
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 8475
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: If the sum of the square roots of two integers is  [#permalink]

### Show Tags

24 Apr 2018, 12:21
1
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 one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only \$79 for 1 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"
Director
Joined: 29 Jun 2017
Posts: 953
Re: If the sum of the square roots of two integers is  [#permalink]

### Show Tags

25 Oct 2018, 00:48
1
how do we know that
x+y=9
and
2 square root of xy= 6 square root of 2
Manager
Status: Quant Expert Q51
Joined: 02 Aug 2014
Posts: 102
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,
_________________
Intern
Joined: 25 Apr 2017
Posts: 14
GMAT 1: 710 Q48 V40
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 : $$a^2-9a+18=0$$

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.
Manager
Joined: 07 Jun 2017
Posts: 100
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=81-2xy
x^2+y^2=81-36=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.
Senior Manager
Status: love the club...
Joined: 24 Mar 2015
Posts: 260
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: 60727
Re: If the sum of the square roots of two integers is  [#permalink]

### Show Tags

01 Oct 2017, 03:59
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 Questions
Roots PS Questions

Hope it helps.
_________________
Senior Manager
Status: love the club...
Joined: 24 Mar 2015
Posts: 260
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 Questions
Roots PS Questions

Hope it helps.

thanks man
great you are
Current Student
Joined: 19 Aug 2016
Posts: 145
Location: India
GMAT 1: 640 Q47 V31
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
Senior SC Moderator
Joined: 22 May 2016
Posts: 3725
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 : $$a^2-9a+18=0$$

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.
_________________
SC Butler has resumed! Get two SC questions to practice, whose links you can find by date, here.

Never doubt that a small group of thoughtful, committed citizens can change the world; indeed, it's the only thing that ever has -- Margaret Mead
Director
Joined: 27 May 2012
Posts: 948
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
=

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
Director
Joined: 27 May 2012
Posts: 948
Re: If the sum of the square roots of two integers is  [#permalink]

### Show Tags

22 Feb 2018, 12:25
niks18 wrote:

Hi stne

Here 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
Re: If the sum of the square roots of two integers is   [#permalink] 22 Feb 2018, 12:25

Go to page    1   2    Next  [ 23 posts ]

Display posts from previous: Sort by

# If the sum of the square roots of two integers is

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne