(rt(9 + rt80) + rt(9 - rt80))^2 =

[GMAT 5/18]

Source: Gmat Prep, mba.com, Test II

This question is a little challenging to post, but hopefully it is clear. I am looking for the fastest and most efficient way to solve this problem. I used estimation when I did it, and that left 2 answers that were very close. So, I had to go back and use more accurate estimation and got the answer (this took a little bit longer than I think this question deserves, so I was hoping for some tips!). Thanks!

(rt(9 + rt80) + rt(9 - rt80))^2 =

a. 1
b. 9 - 4.rt5
c. 18 - 4.rt5
d. 18
e. 20

I will post the answer later.

[StaceyKoprince]

Tough question! If you're going for a top score, make sure you know the three common quadratic "perfect square" equations and how to use them with a weird variation like this. This one's complicated even with the "shortcut."

This starts as a variation of (a+b)^2 = a^2 + 2ab + b^2.
so:
[rt(9 + rt80) + rt(9 - rt80)]^2 =
[rt(9 + rt80)]^2 + [(2)(rt(9 + rt80)rt(9 - rt80)] + [rt(9 - rt80)]^2 Square roots on first and third terms cancel out:
(9 + rt80) + [(2)(rt(9 + rt80)rt(9 - rt80)] + (9 - rt80)
and middle term is now a variation of (a+b)(a-b) = a^2 - b^2:
(9 + rt80) + [(2))rt{(9 + rt80)(9 - rt80)}] + (9 - rt80)
(9 + rt80) + [(2)(81-80)] + (9 - rt80)
9 + 2 + 9 = 20


(Note: this looks even messier on screen than it really is - try writing it out for yourself)

[Guest]

Hi Stacey,

I worked out your method but got stuck somewhere in the middle. Instead of considering the solution as (a+b)^2, why can't we pull out the common "rt" term out at the first stage itself and simplify the terms before starting to square them ? Though I didn't get an answer, listed out in the five options, can you point out where I am going wrong:

Q: ( rt (9+rt80) + rt (9 -rt80) ) ^2

Soln: ( rt [ 9+ rt80 + 9 - rt80 ] ) ^2

= ( rt [ 9 +9 ] ) ^2

= ( rt 18 ) ^2

[Guest]

b^2

what does that mean? I am new to online post?

division or b to the power of 2(exponent 2)

Thanks.

[JadranLee]

It means b squared, i.e. b to the power of 2.

b^2

what does that mean? I am new to online post?

division or b to the power of 2(exponent 2)

Thanks.

[Amit_Puranik]

This looks better on a scratch paper - I promise. :)

[ (9+(80)^1/2)^1/2+ (9-(80)^1/2)^1/2 ] ^ 2

[ ((81)^1/2 + (80)^1/2)^1/2 + ((81)^1/2 - (80)^1/2)^1/2 ] ^ 2

( use the (a+b)^2 formula and (a+b) * (a-b)

((81)^1/2 + (80)^1/2 + (81)^1/2 - (80)^1/2 + 2 [ ((81)^1/2 + (80)^1/2)) * ((81)^1/2 - (80)1/2)) ]

(81)^1/2 + (81) ^1/2 + 2 (81 - 80)

9 + 9 + 2 = 20

ps: I home I have not missed any "()[]"

[mdmdmd63]

I AM AIMING FOR 700+
WHAT ARE THE FORMULAS I NEED TO KNOW
CAN SOM1 TELL?
THANKS

[StaceyKoprince]

There are many formulas you need to know, too many to post in a forum setting. You will need to either check www.mba.com or buy a prep book that lists all of the necessary formulas across the various branches of math tested on the GMAT.

[ryan.m.doyle]

Tough question! If you're going for a top score, make sure you know the three common quadratic "perfect square" equations and how to use them with a weird variation like this. This one's complicated even with the "shortcut."

This starts as a variation of (a+b)^2 = a^2 + 2ab + b^2.
so:
[rt(9 + rt80) + rt(9 - rt80)]^2 =
[rt(9 + rt80)]^2 + [(2)(rt(9 + rt80)rt(9 - rt80)] + [rt(9 - rt80)]^2 Square roots on first and third terms cancel out:
(9 + rt80) + [(2)(rt(9 + rt80)rt(9 - rt80)] + (9 - rt80)
and middle term is now a variation of (a+b)(a-b) = a^2 - b^2:
(9 + rt80) + [(2))rt{(9 + rt80)(9 - rt80)}] + (9 - rt80)
(9 + rt80) + [(2)(81-80)] + (9 - rt80)
9 + 2 + 9 = 20


(Note: this looks even messier on screen than it really is - try writing it out for yourself)

I think there is slight step missing here that was confusing me. Correct me if I'm wrong but from here:
(9 + rt80) + [(2))rt{(9 + rt80)(9 - rt80)}] + (9 - rt80)
to here:
(9 + rt80) + [(2)(81-80)] + (9 - rt80)
You're are dropping the rt over the 81-80...
It should be
(9 + rt80) + [(2)(rt(81-80))] + (9 - rt80)

Which is:
9 +2rt(1) + 9
9 + 2 + 9 = 20

[StaceyKoprince]

yep! I accidentally forgot to retype that "rt" symbol between the 2 and the rest of that mess. :)

[msbinu]

I still didtn get what is wrong with the below approch
( rt (9+rt80) + rt (9 -rt80) ) ^2

Soln: ( rt [ 9+ rt80 + 9 - rt80 ] ) ^2

= ( rt [ 9 +9 ] ) ^2

= ( rt 18 ) ^2

could any one pls explain

[akhp77]

( sqrt( 9 + sqrt( 80 ) ) + sqrt( 9 - sqrt( 80 ) ) ^2
a^2 = 9 + sqrt( 80 )
b^2 = 9 - sqrt( 80 )

a^2 + b^2 = 18

(ab)^2 = 81 - 80 =1

(a+b)^2 = 18 + 2*1 = 20

[StaceyKoprince]

msbinu, you can't combine the two separate roots in the way that you did (from your first to second step). Try this with more normal numbers to see why it doesn't work:

SQRT(16) + SQRT(25) = ?

I gave you perfect squares here so it would be easy to simply each SQRT by itself. We would get 4+5=9, right?

Okay, now try to simplify it in the way that you did above, where you combine 16+25 under the sqrt sign:

SQRT(16+25) =?

Simplify that. Do you still get 9?

Nope. When you have different square roots that are added or subtracted, it isn't okay to just combine them all under one square root sign.