How come the answer is 20 not 18?

[Lookingfor700GMAT]

((square root of 9+sqr80)+square root of 9-sqr80))^2

I thought the answer is 80 as the ^2 cancels the two square roots then I added 9+sqr80 to 9-sqr80 which should be 18 right, as the 9s sum to 18 and the sqr80 cancel out?

[kstv]

((square root of 9+sqr80)+square root of 9-sqr80))^2
reduce it in the form ((a+b)+a-b))²= (a+b+a-b)²= (2a)² = 4(sq rt 9)²= 4*9
are the bracket placement same as above?

[Lookingfor700GMAT]

But the answer is 20 on the GMAT prep not 36 (4*9)

[Brian@VeritasPrep]

This is a great question - one of my students brought this in to class the other day...he scored something like 750 on his practice test and this was the lone question that was still driving him nuts afterward.

It's ugly in text form without the ability to draw radical signs, etc. but essentially the problem will break out this way:

1) Square the initial statement (use FOIL if you don't feel overly comfortable with the memorization for (x+y)^2, to get:

First term: 9 + sqrt80 + Second term: sqrt(9 + sqrt80) * sqrt(9 - sqrt 80) + Third term: sqrt (9 + sqrt 80) * sqrt(9 - sqrt 80) + Last term: 9 - sqrt 80

2) Clean up what you have - notice that you have a +sqrt 80 and a -sqrt 80, so those will cancel out to zero, and you have 9+9, which gives you 18. The middle terms are the same, so you can add them together to get:

18 + 2 sqrt (9 + sqrt80) * (sqrt (9 - sqrt 80)

3) Hopefully you've noticed by now that you're really close to being able use the "magic" Difference of Squares, (x+y)(x-y) = x^2 - y^2. The GMAT will reward you for knowing that one - it's worth memorizing and looking for. If that's your guide, the radical rules will come a bit easier. You can note that sqrt x * sqrt y = sqrt (xy), which would allow you to multiply together the outer radicals in the statement above:

18 + 2squrt ((9 + sqrt 80)(9 - sqrt 80))

4) That lets you apply that difference of squares rule nicely. 9^2 is 81, and (sqrt 80)^2 is 80, giving you:

18+2sart (81-80)

18 + 2 sqrt 1

18 + 2 = 20


The biggest keys to this one are to:

-Stay patient and stay organized. The question looks a lot uglier than it is.

-Know the GMAT and its tendencies. That "Difference of Squares" rule is one of the ultimate "decoder" tools you have at your disposal. If you see that you have an x+y and an x-y term somewhere in a difficult problem, it's pretty likely that you'll need to get them together to simplify. When my ace student, Andy, brought this in to class the other night, I looked at it and immediately told him we'd eventually use Difference of Squares...it was only a matter of simplifying the algebra to get there.

-Look at the answer choices - they're all integers on this one if I recall correctly, so that should guide you, too. You know you can get there...just stay patient and organized.

Great question - thanks for bringing this one up! I have pretty good evidence that this one is a very difficult question (a student scoring in the 700s got and missed this one), so if you're nodding along as you read this, you're poised to do well!

[kstv]

Are there others like me who find readability of a post like 2^5*7 + 3^5 not very good.

One way is to use ASCII codes. alt 252 for ² alt 248 for degrees ° etc
But remembering them is difficult.
Like "Add Emotions Tab "why not have a tab for the commonly used Maths Symbols .
Is it feasible ?

[Phirozz]

But the answer is 20 on the GMAT prep not 36 (4*9)

Could u plz restate the question.. still I m unable to get

[aerodan1]

i too am finding this problem hard to work out because of the confusion stemming from the readability, or lack thereof. if im correct, this is essentially an (a + b)^2 problem. a=(sq. rt. of 9 + sq. rt. of 80) and b=(sq. rt. of 9 - sq. rt. of 80). THANKS. - dan

[AtifS]

Actually I think it's
[{sqrt(9+sqrt80)} + {sqrt(9-sqrt80)}]^2
Will be solved using following basic formula--> (a+b)^2= a^2 + b^2 + 2ab
so now let's solve it
{sqrt(9+sqrt80) + sqrt(9-sqrt80)}^2
={sqrt(9+sqrt80)}^2 + {sqrt(9-sqrt80)}^2 +2 * {sqrt(9+sqrt80)}*{sqrt(9-sqrt80)}

sqrt and ^2 (sqaure or power 2) will cancel each other and by solving for bold terms using formula (a+b)*(a-b)=a^2 - b^2 and at the same time merging them into one sqrt as they are multiplying terms, we get

=(9+sqrt80) + (9-sqrt80) + 2 * sqrt{(9)^2 - (sqrt80)^2}
By opening the brackets of first two terms and in last term canceling sqrt & ^2 like before, we get
=9 + sqrt80 + 9 - sqrt80 + 2 * sqrt{81- 80}
+sqrt80 & -sqrt80 will cancel each other and both 9's will add up, so
=18+ 2* sqrt{1}=18+2*(1)=18+2= 20 -->Answer
Hence equation results 20 as an answer.
Hope it helps!

Actually, question wasn't written properly, that's why people had difficulty understanding the question.

[kstv]

trying the same method of substituting the value
[ √(9 + √80) + √(9 - √80) }² a = 9 b = √80
[(√(a+b)+√(a-b)]²= a+b+a-b+2√(a+b)(a-b) = 2a + 2√a²-b²
2*9 + 2√81-80 = 18 + 2 = 20
Phew ! Much simpler
thanks to AtifS

[aerodan1]

yes, its true that the question was written incorrectly, and now that i realize what the question is, i see the logic behind brians post. knowing the radical rules is just as important as recognizing the difference of squares rule, in fact, more vital to solving this problem i think.

[AtifS]

trying the same method of substituting the value
[ √(9 + √80) + √(9 - √80) }² a = 9 b = √80
[(√(a+b)+√(a-b)]²= a+b+a-b+2√(a+b)(a-b) = 2a + 2√a²-b²
2*9 + 2√81-80 = 18 + 2 = 20
Phew ! Much simpler
thanks to AtifS

Thank you! for explaining it in an easy way as well as using sqrt symbols (I wanted to use but couldn't find any facility in posting options). Did you solve it on some separate doc or software (e.g MS Word) and then pasted it here. By using symbols makes it easy.

[kstv]

press the ALT key the type the no 251 u will get √
cos ASCII code for √ is 251
try it
This feature works on BTG now.

[prose]

Wow, what a beast of a problem; before looking at the explanations I thought I was going crazy. It all makes sense now though, especially since I initially misunderstood the way the problem was written out. After awhile I realized that it was in (a+b)^2 format. That helped out tremendously, as well as the other explanations in this thread. Thanks!

[sanju09]

Are there others like me who find readability of a post like 2^5*7 + 3^5 not very good.

One way is to use ASCII codes. alt 252 for ² alt 248 for degrees ° etc
But remembering them is difficult.
Like "Add Emotions Tab "why not have a tab for the commonly used Maths Symbols .
Is it feasible ?

What an Idea Sir G!! I would request you to post as many ASCII codes as you could, on a separate thread, which could be considered as STICKY, and keep updating it. I have already made similar suggestions to the forum heads and I am sure that they would very soon come up with some cool alternative for this too; but till then, your ASCII codes thread would ROCK