Last visit was: 25 Apr 2026, 03:19 It is currently 25 Apr 2026, 03:19
Home
Messages
Tests
Schools
Events
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.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
User avatar
capable2
User avatar
Joined: 23 Oct 2009
Last visit: 12 Nov 2009
Posts: 3
Given Kudos: 1
Posts: 3
Kudos: 0
OG 12th edition sqrt problem 12 Nov 2009, 06:04
Kudos
Add Kudos
Bookmarks
Bookmark this Post
#32 on page 156 has a rather simple sqrt problem, which I am able to solve using a brute force method for finding the sqrt. However, I'd like to understand the book's solution for finding the solution more quickly, but don't quite follow it. Any help would be appreciated.

I've attached an image showing the book's solution. I'm wondering how I am able to drop the 16 as one of the root multiples of 256 (the product of 8*32).

Appreciate any help!



Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Quantitative Questions Forum
Still interested in this question? Check out the "Best Topics" block below for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 25 Apr 2026
Posts: 109,822
Own Kudos:
811,135
Given Kudos: 105,878
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,822
Kudos: 811,135
OG 12th edition sqrt problem 12 Nov 2009, 06:17
Kudos
Add Kudos
Bookmarks
Bookmark this Post
The solution in book is quite strange. Generally \(\sqrt{a+b}\), DOES NOT equal to \(\sqrt{a}+\sqrt{b}\). Example: \(\sqrt{16+16}=\sqrt{32}=4*\sqrt{2}\) and NOT \(\sqrt{16}+\sqrt{16}=4+4=8\).

Back to the question:

As I understand the problem is:

\(\sqrt{16*20+8*32}\) if yes then: \(\sqrt{16*(20+8*2)}=\sqrt{16*36}=4*6=24\)

Hope it helps.
User avatar
capable2
User avatar
Joined: 23 Oct 2009
Last visit: 12 Nov 2009
Posts: 3
Given Kudos: 1
Posts: 3
Kudos: 0
OG 12th edition sqrt problem 12 Nov 2009, 08:24
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Argh. Sorry, there is a typo in my image - you are correct that it should be \(\sqrt{16}*\sqrt{36}\) and not \(\sqrt{16}+\sqrt{36}\).

I am embarrassed that I still do not understand how (8*32) becomes (8*2)? 8*32 is 256, the root of which is 16. So by my thinking, that would leave me with \(\sqrt{(16)(20)+(16)(16)}\). How do you get \(\sqrt{(16)(20)+(8)(2)}\) from that?

Thanks again.
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 25 Apr 2026
Posts: 109,822
Own Kudos:
811,135
 [1]
Given Kudos: 105,878
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,822
Kudos: 811,135
 [1]
OG 12th edition sqrt problem 12 Nov 2009, 08:42
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
capable2
Argh. Sorry, there is a typo in my image - you are correct that it should be \(\sqrt{16}*\sqrt{36}\) and not \(\sqrt{16}+\sqrt{36}\).

I am embarrassed that I still do not understand how (8*32) becomes (8*2)? 8*32 is 256, the root of which is 16. So by my thinking, that would leave me with \(\sqrt{(16)(20)+(16)(16)}\). How do you get \(\sqrt{(16)(20)+(8)(2)}\) from that?

Thanks again.
Show more

The red part is not correct we'll get not the expression you wrote but: \(\sqrt{16*(20+8*2)}\). Look at the brackets.

And here is how:

Let's just forget about the square root for a moment. We have:

\(16*20+8*32\) no need to multiply \(8\) by \(32\), it's a long way, though still correct. Just take \(16\) from the brackets:

\(16*20+8*32=16*20+8*2*16=16*(20+8*2)=16*(20+16)=16*36\).

Back to the root. We'll have \(\sqrt{16*36}=4*6=24\)

Generally \(\sqrt{a*b}=\sqrt{a}*\sqrt{b}\)

Hope it's clear.
avatar
joshuaRome
avatar
Joined: 13 Jan 2012
Last visit: 19 Jun 2012
Posts: 2
Posts: 2
Kudos: 0
OG 12th edition sqrt problem 29 Jan 2012, 21:15
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hello everyone,

I am having some difficulty with this problem similar to the author of this post.

My problem (and possibly flawed logic):

It is my understanding that \sqrt{2}+\sqrt{2}=2\sqrt{2}
and that adding square roots produces the wrong answer.

to say that \sqrt{2}+\sqrt{2}=\sqrt{4}=2 is incorrect from what i see. (1.4appx+1.4appx is not 2)


problem 32 OG:

sqrt{ 16 (20 + 8 * 2)}

sqrt{ 16 (20 + 16)}
okay we factored out 16, makes sense to me...

sqrt{16 (36)}

i don't see how we can add two square roots together. square root of 20 + square root 16 is 8.47 not 6.

Thank you for any responses! I know I must have made a mistake but sometimes a different perspective can help.
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 25 Apr 2026
Posts: 109,822
Own Kudos:
811,135
Given Kudos: 105,878
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,822
Kudos: 811,135
OG 12th edition sqrt problem 30 Jan 2012, 02:23
Kudos
Add Kudos
Bookmarks
Bookmark this Post
joshuaRome
Hello everyone,

I am having some difficulty with this problem similar to the author of this post.

My problem (and possibly flawed logic):

It is my understanding that \sqrt{2}+\sqrt{2}=2\sqrt{2}
and that adding square roots produces the wrong answer.

to say that \sqrt{2}+\sqrt{2}=\sqrt{4}=2 is incorrect from what i see. (1.4appx+1.4appx is not 2)


problem 32 OG:

sqrt{ 16 (20 + 8 * 2)}

sqrt{ 16 (20 + 16)}
okay we factored out 16, makes sense to me...

sqrt{16 (36)}

i don't see how we can add two square roots together. square root of 20 + square root 16 is 8.47 not 6.

Thank you for any responses! I know I must have made a mistake but sometimes a different perspective can help.
Show more

The point is that we are not adding square root of 20 and square root of 16, we are adding 20 and 16 under the SAME square root. Below it he solution to the problem from OG:

\(\sqrt{16*20+8*32}=?\) --> factor out 16: \(\sqrt{16(20+8*2)}=\sqrt{16*36}=4*6=24\).

For more on this topic check Number theory chapter of Math Book (part about roots): math-number-theory-88376.html

Tough problems on exponents and roots (with detailed solutions) to practice:
PS: tough-and-tricky-exponents-and-roots-questions-125956.html
DS: tough-and-tricky-exponents-and-roots-questions-125967.html

Hope it helps.
User avatar
KarishmaB
User avatar
Joined: 16 Oct 2010
Last visit: 23 Apr 2026
Posts: 16,442
Own Kudos:
79,407
Given Kudos: 485
Location: Pune, India
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 16,442
Kudos: 79,407
OG 12th edition sqrt problem 30 Jan 2012, 02:52
Kudos
Add Kudos
Bookmarks
Bookmark this Post
joshuaRome


sqrt{ 16 (20 + 16)}
okay we factored out 16, makes sense to me...

sqrt{16 (36)}

i don't see how we can add two square roots together. square root of 20 + square root 16 is 8.47 not 6.

Thank you for any responses! I know I must have made a mistake but sometimes a different perspective can help.
Show more

Can we write\(\sqrt{4}\) as \(\sqrt{2+2}\)? Sure.
But \(\sqrt{2+2} \neq \sqrt{2} + \sqrt{2}\)

If addition/subtraction are under the square root, it is fine. But you cannot separate the added terms and cannot give them separate square roots.

Therefore, \(\sqrt{16+20}\) = \(\sqrt{36}\)

But, \(\sqrt{16+20} \neq \sqrt{16} + \sqrt{20}\)

For basic theory of roots, check out: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/08 ... -the-gmat/
User avatar
Bigred2008
User avatar
Joined: 10 Aug 2009
Last visit: 08 Mar 2017
Posts: 66
Own Kudos:
294
Given Kudos: 45
Location: United States (VA)
Concentration: Entrepreneurship, Strategy
Schools: Darden '17 (M$)
WE:Business Development (Other)
Products:
My Reviews
Schools: Darden '17 (M$)
Posts: 66
Kudos: 294
OG 12th edition sqrt problem 15 Nov 2014, 16:45
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
joshuaRome
Hello everyone,

I am having some difficulty with this problem similar to the author of this post.

My problem (and possibly flawed logic):

It is my understanding that \sqrt{2}+\sqrt{2}=2\sqrt{2}
and that adding square roots produces the wrong answer.

to say that \sqrt{2}+\sqrt{2}=\sqrt{4}=2 is incorrect from what i see. (1.4appx+1.4appx is not 2)


problem 32 OG:

sqrt{ 16 (20 + 8 * 2)}

sqrt{ 16 (20 + 16)}
okay we factored out 16, makes sense to me...

sqrt{16 (36)}

i don't see how we can add two square roots together. square root of 20 + square root 16 is 8.47 not 6.

Thank you for any responses! I know I must have made a mistake but sometimes a different perspective can help.

The point is that we are not adding square root of 20 and square root of 16, we are adding 20 and 16 under the SAME square root. Below it he solution to the problem from OG:

\(\sqrt{16*20+8*32}=?\) --> factor out 16: \(\sqrt{16(20+8*2)}=\sqrt{16*36}=4*6=24\).

For more on this topic check Number theory chapter of Math Book (part about roots): math-number-theory-88376.html

Tough problems on exponents and roots (with detailed solutions) to practice:
PS: tough-and-tricky-exponents-and-roots-questions-125956.html
DS: tough-and-tricky-exponents-and-roots-questions-125967.html

Hope it helps.
Show more

I just have one question, wouldn't there still be a 1 left over when you factor out the 16? I'm just thinking if you were to reverse the operation, what would the 16 get multiplied into?
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 25 Apr 2026
Posts: 109,822
Own Kudos:
811,135
Given Kudos: 105,878
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,822
Kudos: 811,135
OG 12th edition sqrt problem 17 Nov 2014, 12:23
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bigred2008
Bunuel
joshuaRome
Hello everyone,

I am having some difficulty with this problem similar to the author of this post.

My problem (and possibly flawed logic):

It is my understanding that \sqrt{2}+\sqrt{2}=2\sqrt{2}
and that adding square roots produces the wrong answer.

to say that \sqrt{2}+\sqrt{2}=\sqrt{4}=2 is incorrect from what i see. (1.4appx+1.4appx is not 2)


problem 32 OG:

sqrt{ 16 (20 + 8 * 2)}

sqrt{ 16 (20 + 16)}
okay we factored out 16, makes sense to me...

sqrt{16 (36)}

i don't see how we can add two square roots together. square root of 20 + square root 16 is 8.47 not 6.

Thank you for any responses! I know I must have made a mistake but sometimes a different perspective can help.

The point is that we are not adding square root of 20 and square root of 16, we are adding 20 and 16 under the SAME square root. Below it he solution to the problem from OG:

\(\sqrt{16*20+8*32}=?\) --> factor out 16: \(\sqrt{16(20+8*2)}=\sqrt{16*36}=4*6=24\).

For more on this topic check Number theory chapter of Math Book (part about roots): math-number-theory-88376.html

Tough problems on exponents and roots (with detailed solutions) to practice:
PS: tough-and-tricky-exponents-and-roots-questions-125956.html
DS: tough-and-tricky-exponents-and-roots-questions-125967.html

Hope it helps.

I just have one question, wouldn't there still be a 1 left over when you factor out the 16? I'm just thinking if you were to reverse the operation, what would the 16 get multiplied into?
Show more

You mean it should be \(\sqrt{16(20*1+8*2)}=\sqrt{16*36}=4*6=24\)? Yes, but 20*1 = 20, so we can omit 1 there.

This question is discussed here: root-of-138296.html

Hope it helps.



Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Quantitative Questions Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!