Last visit was: 19 Jul 2025, 14:31 It is currently 19 Jul 2025, 14:31
Close
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
Your Progress

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
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Close
Request Expert Reply
Confirm Cancel
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 19 July 2025
Posts: 102,625
Own Kudos:
Given Kudos: 98,235
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,625
Kudos: 742,779
 [50]
Kudos
Add Kudos
50
Bookmarks
Bookmark this Post
Most Helpful Reply
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 19 July 2025
Posts: 102,625
Own Kudos:
Given Kudos: 98,235
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,625
Kudos: 742,779
 [20]
6
Kudos
Add Kudos
13
Bookmarks
Bookmark this Post
avatar
AlexBunea
Joined: 25 Jul 2016
Last visit: 10 Oct 2018
Posts: 4
Own Kudos:
19
 [8]
Given Kudos: 16
GMAT 1: 740 Q50 V40
GMAT 1: 740 Q50 V40
Posts: 4
Kudos: 19
 [8]
5
Kudos
Add Kudos
3
Bookmarks
Bookmark this Post
General Discussion
avatar
pkappaga
Joined: 10 Nov 2015
Last visit: 25 Feb 2018
Posts: 2
Own Kudos:
1
 [1]
Given Kudos: 1
Posts: 2
Kudos: 1
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
I understand the simplification part but Isn`t the final value 60? I am not sure why you are doing the square root of 60
User avatar
ENGRTOMBA2018
Joined: 20 Mar 2014
Last visit: 01 Dec 2021
Posts: 2,328
Own Kudos:
3,791
 [3]
Given Kudos: 816
Concentration: Finance, Strategy
GMAT 1: 750 Q49 V44
GPA: 3.7
WE:Engineering (Aerospace and Defense)
Products:
GMAT 1: 750 Q49 V44
Posts: 2,328
Kudos: 3,791
 [3]
2
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
pkappaga
I understand the simplification part but Isn`t the final value 60? I am not sure why you are doing the square root of 60

It is done becaue Bunuel starts with a squared value of the expression that we need to evalute (look below). This is the reason why you then have to take the square root to get the value of the question asked.


You need to find the value of \((\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}})\) and NOT \((\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}})^2\)

Bunuel

Square the given expression to get rid of the roots, though don't forget to un-square the value you get at the end to balance this operation and obtain the right answer:

Must know for the GMAT: \((x+y)^2=x^2+2xy+y^2\) (while \((x-y)^2=x^2-2xy+y^2\)).

So we get:

\((\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}})^2 =\) \(=(\sqrt{25+10\sqrt{6}})^2+2(\sqrt{25+10\sqrt{6}})(\sqrt{25-10\sqrt{6}})+(\sqrt{25-10\sqrt{6}})^2=\)


Hope this helps.
User avatar
akela
Joined: 30 Jan 2016
Last visit: 23 May 2023
Posts: 1,228
Own Kudos:
5,665
 [6]
Given Kudos: 128
Products:
Posts: 1,228
Kudos: 5,665
 [6]
6
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Alternative solution: 10*\(\sqrt{6}\) = \(\sqrt{600}\), which is approximately 24.5 as 600 lies between 24*24=576 and 25*25=625
So, \(\sqrt{25+24.5}\) + \(\sqrt{25-24.5}\) = \(\sqrt{49,5}\) + \(\sqrt{0.5}\) ≈ slightly more than 7+ around 0.7*≈ 8

* 0.7*0.7 =0.49 ≈ 0,5

So, we can eliminate A, D and E

B and C
2*\(\sqrt{15}\)≈ 2*4 ≈ 8 - contender

\(\sqrt{55}\) - loser.
First, 55 is closer to 49 than to 64, so \(\sqrt{55}\) is closer to 7.
Second, using guessing technique, we see that 55 has no pair with any other number in answer choices, whilst there is a pair of 60s - 60 and \(\sqrt{60}\).

Hope this helps!
avatar
JavierS
avatar
Current Student
Joined: 23 Nov 2015
Last visit: 19 Jun 2018
Posts: 9
Own Kudos:
21
 [7]
Given Kudos: 74
GMAT 1: 690 Q45 V39
GPA: 3.47
Products:
GMAT 1: 690 Q45 V39
Posts: 9
Kudos: 21
 [7]
5
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
Bunuel
Official Solution:

What is the value of \(\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}}\)?

A. \(2\sqrt{5}\)
B. \(\sqrt{55}\)
C. \(2\sqrt{15}\)
D. 50
E. 60


Square the given expression to get rid of the roots, though don't forget to un-square the value you get at the end to balance this operation and obtain the right answer:

Must know for the GMAT: \((x+y)^2=x^2+2xy+y^2\) (while \((x-y)^2=x^2-2xy+y^2\)).

So we get:

\((\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}})^2 =\) \(=(\sqrt{25+10\sqrt{6}})^2+2(\sqrt{25+10\sqrt{6}})(\sqrt{25-10\sqrt{6}})+(\sqrt{25-10\sqrt{6}})^2=\)

\(=(25+10\sqrt{6})+2(\sqrt{25+10\sqrt{6}})(\sqrt{25-10\sqrt{6}})+(25-10\sqrt{6})\).

Note that sum of the first and the third terms simplifies to \((25+10\sqrt{6})+(25-10\sqrt{6})=50\), so we have

\(50+2(\sqrt{25+10\sqrt{6}})(\sqrt{25-10\sqrt{6}})\) therefore:

\(50+2(\sqrt{25+10\sqrt{6}})(\sqrt{25-10\sqrt{6}})=\) \(50+2\sqrt{(25+10\sqrt{6})(25-10\sqrt{6})}\).

Also must know for the GMAT: \((x+y)(x-y)=x^2-y^2\), thus

\(50+2\sqrt{(25+10\sqrt{6})(25-10\sqrt{6})}=50+2\sqrt{25^2-(10\sqrt{6})^2)} =\) \(= 50+2\sqrt{625-600}=50+2\sqrt{25}=60\).

Recall that we should un-square this value to get the right answer: \(\sqrt{60}=2\sqrt{15}\).


Answer: C

Can it be solved the following way?
\(\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}}=\sqrt{(\sqrt{15}+\sqrt{10})^2}+\sqrt{(\sqrt{15}-\sqrt{10})^2}\)

then it would turn into \(|\sqrt{15}+\sqrt{10}| +|\sqrt{15}-\sqrt{10}|\)
then \(\sqrt{15}+\sqrt{10}+\sqrt{15}-\sqrt{10}=2\sqrt{15}\)
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 19 July 2025
Posts: 102,625
Own Kudos:
742,779
 [1]
Given Kudos: 98,235
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,625
Kudos: 742,779
 [1]
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
MikeMighty
Bunuel
Official Solution:

What is the value of \(\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}}\)?

A. \(2\sqrt{5}\)
B. \(\sqrt{55}\)
C. \(2\sqrt{15}\)
D. 50
E. 60


Square the given expression to get rid of the roots, though don't forget to un-square the value you get at the end to balance this operation and obtain the right answer:

Must know for the GMAT: \((x+y)^2=x^2+2xy+y^2\) (while \((x-y)^2=x^2-2xy+y^2\)).

So we get:

\((\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}})^2 =\) \(=(\sqrt{25+10\sqrt{6}})^2+2(\sqrt{25+10\sqrt{6}})(\sqrt{25-10\sqrt{6}})+(\sqrt{25-10\sqrt{6}})^2=\)

\(=(25+10\sqrt{6})+2(\sqrt{25+10\sqrt{6}})(\sqrt{25-10\sqrt{6}})+(25-10\sqrt{6})\).

Note that sum of the first and the third terms simplifies to \((25+10\sqrt{6})+(25-10\sqrt{6})=50\), so we have

\(50+2(\sqrt{25+10\sqrt{6}})(\sqrt{25-10\sqrt{6}})\) therefore:

\(50+2(\sqrt{25+10\sqrt{6}})(\sqrt{25-10\sqrt{6}})=\) \(50+2\sqrt{(25+10\sqrt{6})(25-10\sqrt{6})}\).

Also must know for the GMAT: \((x+y)(x-y)=x^2-y^2\), thus

\(50+2\sqrt{(25+10\sqrt{6})(25-10\sqrt{6})}=50+2\sqrt{25^2-(10\sqrt{6})^2)} =\) \(= 50+2\sqrt{625-600}=50+2\sqrt{25}=60\).

Recall that we should un-square this value to get the right answer: \(\sqrt{60}=2\sqrt{15}\).


Answer: C

Can it be solved the following way?
\(\sqrt{25+10\sqrt{6}}+\sqrt{25-10\sqrt{6}}=\sqrt{(\sqrt{15}+\sqrt{10})^2}+\sqrt{(\sqrt{15}-\sqrt{10})^2}\)

then it would turn into \(|\sqrt{15}+\sqrt{10}| +|\sqrt{15}-\sqrt{10}|\)
then \(\sqrt{15}+\sqrt{10}+\sqrt{15}-\sqrt{10}=2\sqrt{15}\)
_______________
Yes, that's correct.
avatar
Milano2017
Joined: 03 Jun 2017
Last visit: 01 Oct 2017
Posts: 6
Own Kudos:
3
 [3]
Given Kudos: 7
Posts: 6
Kudos: 3
 [3]
3
Kudos
Add Kudos
Bookmarks
Bookmark this Post
I find the official solution to be very long and it would take me probably 4-5 minutes to finish the answer. By then I would probably pick 60 since i would forget to take the square root.

My approach was using the approximate numbers for each. First I recognized that sqroot of 6 is 2.5 so the second part was equal to 0. The first part was 25+25=50. Square root of 50 is just over 7. Then I looked that answers and found the number closest to the one I got. It will take you probably less than 1:30 to solve using this method.
avatar
illthinker
Joined: 17 Sep 2017
Last visit: 28 Aug 2019
Posts: 6
Own Kudos:
36
 [4]
Given Kudos: 5
Concentration: Finance, Entrepreneurship
GMAT 1: 730 Q48 V42
GPA: 3.8
GMAT 1: 730 Q48 V42
Posts: 6
Kudos: 36
 [4]
2
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
Let \(25 + 10\sqrt{6}\ = A\) &
Let \(25 - 10\sqrt{6}\ = B =>\)

\(\sqrt{A}\ + \sqrt{B}\ = x\)
\(A + B + 2\sqrt{AB}\ = x^2 =>\)
\(25 + 10\sqrt{6}\ + 25 - 10\sqrt{6}\ + 2\sqrt{(25 + 10\sqrt{6})(25 - 10\sqrt{6})}\ = x^2\)
\(50 + 2\sqrt{25^2 - (10\sqrt{6})^2}\ = x^2 =>\)
\(x^2 = 60\)
\(x = 2\sqrt{15}\)
User avatar
energetics
Joined: 05 Feb 2018
Last visit: 09 Oct 2020
Posts: 298
Own Kudos:
906
 [3]
Given Kudos: 325
Posts: 298
Kudos: 906
 [3]
3
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Imo, it's much easier/faster to estimate, especially since D, E are so far apart from A,B,C:
I knew that √5 ≈ 2.25 so let's call √6 ≈ 2.4 (actual is 2.449...)

\(\sqrt{25+10*2.4}+\sqrt{25-10*2.4}\)
\(\sqrt{25+24}+\sqrt{25-24}\)
\(\sqrt{49}+\sqrt{1}\)
\(7 + 1\)
So our answer should be close to 8, D,E are out immediately
C can be rewritten as \(\sqrt{60}\) which is closer than B \(\sqrt{55}\) , so it's C.
User avatar
Pari28
Joined: 24 Feb 2014
Last visit: 19 Dec 2019
Posts: 33
Own Kudos:
Given Kudos: 895
Location: United States (GA)
WE:Information Technology (Computer Software)
Posts: 33
Kudos: 9
Kudos
Add Kudos
Bookmarks
Bookmark this Post
I think this is a high-quality question and I agree with explanation.
avatar
sarthak952
Joined: 16 Feb 2020
Last visit: 10 Sep 2022
Posts: 9
Own Kudos:
Given Kudos: 249
Posts: 9
Kudos: 6
Kudos
Add Kudos
Bookmarks
Bookmark this Post
I think this is a high-quality question and I agree with explanation.
avatar
danklin
Joined: 29 Jul 2020
Last visit: 23 Aug 2022
Posts: 9
Own Kudos:
Given Kudos: 829
Posts: 9
Kudos: 2
Kudos
Add Kudos
Bookmarks
Bookmark this Post
I think this is a high-quality question and I agree with explanation.
User avatar
rickyric395
Joined: 07 Mar 2020
Last visit: 18 Jul 2025
Posts: 113
Own Kudos:
Given Kudos: 55
GMAT 1: 680 Q49 V34
GMAT 1: 680 Q49 V34
Posts: 113
Kudos: 92
Kudos
Add Kudos
Bookmarks
Bookmark this Post
My 2 cents : I guess since options are spread pretty far apart except B and C, We can try approximating the answer
\(\sqrt{6}\) is approx 2.4 and \(10\sqrt{6}\) is approx 24. so 1st sqr root can be approximate to \(\sqrt{25+24}\) i.e aprrox \(\sqrt{49}\) i.e 7 and 2nd sqr root can be approximate to \(1\) similarly. Total is \(8\) (approx).

option A : \(2\sqrt{5}\) is \(2*2.2\) = 4.4
option B : \(\sqrt{55}\) is between 7 and 8 as \(\sqrt{49}\) is 7 and \(\sqrt{64}\) is 8 but since 55 is closer to 49 than 64 so value is less than 7.5.
option C : \(2*\sqrt{15}\) , \(\sqrt{15}\) pretty close to 4 , and hence 2*4 =8 . Our required answer.
option D and E can be ignored as they are pretty far away from required answer.
User avatar
lotif
Joined: 16 Jan 2019
Last visit: 19 Apr 2024
Posts: 18
Own Kudos:
Given Kudos: 27
Products:
Posts: 18
Kudos: 4
Kudos
Add Kudos
Bookmarks
Bookmark this Post
In the exam how is one meant to know this is the approach to take? It is not one that for me was intuitive, to take the square then unsquare in the end. If one takes the wrong approach, then you will hve to move on as not enough time to take more than 1 approach. How cn oyu know by just looking at it to do this?
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 19 July 2025
Posts: 102,625
Own Kudos:
742,779
 [1]
Given Kudos: 98,235
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,625
Kudos: 742,779
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
rezalotif
In the exam how is one meant to know this is the approach to take? It is not one that for me was intuitive, to take the square then unsquare in the end. If one takes the wrong approach, then you will hve to move on as not enough time to take more than 1 approach. How cn oyu know by just looking at it to do this?

Squaring an expression containing square roots is a common algebraic technique used to eliminate the square roots and simplify the expression. Conversely, taking the square root is an operation performed to balance the initial squaring. Gaining proficiency in these techniques often comes with practice.
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 19 July 2025
Posts: 102,625
Own Kudos:
Given Kudos: 98,235
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,625
Kudos: 742,779
Kudos
Add Kudos
Bookmarks
Bookmark this Post
I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
User avatar
BottomJee
User avatar
Retired Moderator
Joined: 05 May 2019
Last visit: 09 Jun 2025
Posts: 996
Own Kudos:
Given Kudos: 1,009
Affiliations: GMAT Club
Location: India
GMAT Focus 1: 645 Q82 V81 DI82
GMAT 1: 430 Q31 V19
GMAT 2: 570 Q44 V25
GMAT 3: 660 Q48 V33
GPA: 3.26
WE:Engineering (Manufacturing)
Products:
GMAT Focus 1: 645 Q82 V81 DI82
GMAT 3: 660 Q48 V33
Posts: 996
Kudos: 1,217
Kudos
Add Kudos
Bookmarks
Bookmark this Post
I think this is a high-quality question and I agree with explanation.
User avatar
Vikramaditya00
Joined: 24 Dec 2022
Last visit: 20 Oct 2024
Posts: 47
Own Kudos:
Given Kudos: 9
Posts: 47
Kudos: 1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
I think this is a high-quality question and I agree with explanation. I just didnt get the last part why did we took root 60
 1   2   
Moderators:
Math Expert
102625 posts
Founder
41115 posts