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PERFECT ALGBRA

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PERFECT ALGBRA

by emiflo » Mon Aug 09, 2010 4:44 am
sq root(9+ sq root 80) + sq root( 9- sq root 80) equals

a. 1
b. 9- 4 sq root 5
c. 18- 4 sq root 5
d. 18
e. 20

Hope the question is understandable. I'm trying to say that sq root = ^1/2. hope someone can solve this ASAP.Thanks

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by selango » Mon Aug 09, 2010 8:25 am
emiflo,

Can you please check the question again?

Is it sq root(9+ sq root 80) * sq root( 9- sq root 80)

sq root(9+ sq root 80) * sq root( 9- sq root 80)

=sqrt[(9+ sq root 80)*(9- sq root 80)]

=sqrt[81-80]=sqrt[1]=1

Pick A
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by indiantiger » Mon Aug 09, 2010 9:13 am
The question does not seem right.
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by Brian@VeritasPrep » Mon Aug 09, 2010 9:15 am
Hey guys,

If I'm reading it correctly, this problem comes from one of the official practice tests, and should be written as:

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

If that's the one (and it's one of my favorites - I had a student bring it into class the first time I saw it and put me on the spot), we had a great discussion about it here:

https://www.beatthegmat.com/how-come-the ... tml#235872

As selango predicts based on his interpretation of the question, this one requires you to use that Difference of Squares rule:

(x + y)(x - y) = x^2 - y^2

Whenever you see addition/subtraction of squares or roots, look for an opportunity to use Difference of Squares, which may be the single-most useful algebraic rule on this test!
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by selango » Mon Aug 09, 2010 9:21 am
Thanks Brian.

So if the question is [sq root(9+ sq root 80) + sq root( 9- sq root 80)]^2,the the solution is as below,

9+sqrt(80)+2(1)+9-sqrt(80)=18+2=20
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by Brian@VeritasPrep » Mon Aug 09, 2010 10:04 am
Hey Selango,

You're right...but there are a few steps in the middle to get to that point!

Personally, I'd use that FOIL method to take care of squaring the initial expression (there are enough individual terms in there that I'd rather do it myself than rely on having memorized that (x+y)^2 setup):

[sqrt(9 + sqrt 80) + sqrt (9 - sqrt 80)] * [sqrt(9 + sqrt 80) + sqrt (9 - sqrt 80)]

First Outside Inside Last
9 + sqrt80 + (9+sqrt80)(9-sqrt80) + (9+sqrt80)(9-sqrt80) + 9 - sqrt 80

Then you can start to simplify. you're adding two 9s and you have a +sqrt 80 and a -sqrt 80, so those will cancel so that you have:

18 + (9+sqrt80)(9-sqrt80) + (9+sqrt80)(9-sqrt80)

and since that parenthetical term is replicated, we can just multiply it by 2:

18 + 2(9+sqrt80)(9-sqrt80)

Here you can use that difference of squares rule, which nicely removes the radicals around the square roots:

(9 + sqrt 80)(9 - sqrt 80) = 81 - 80 = 1

So now you have 18 + 2(1) = 20


The algebra looks pretty involved, especially if you have to type it, but to me the key is recognizing that you can use Difference of Squares in the end - if you know where the algebra is leading you, you can pretty confidently go through each step and know that it will work out. When I first saw this problem in class with 20 eyes on me, I wrote down "Difference of Squares" right away to show that I knew where I was going with it...it was only a matter of setting it up to get there.
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by Brent@GMATPrepNow » Mon Aug 09, 2010 12:11 pm
emiflo wrote:sq root(9+ sq root 80) + sq root( 9- sq root 80) equals

a. 1
b. 9- 4 sq root 5
c. 18- 4 sq root 5
d. 18
e. 20

Hope the question is understandable. I'm trying to say that sq root = ^1/2. hope someone can solve this ASAP.Thanks
I like the solutions I've seen so far, except there seems to be a problem.
It seems that we want to find the value of sq root(9+ sq root 80) + sq root( 9- sq root 80)

So, we'll let sq root(9+ sq root 80) + sq root( 9- sq root 80) = k and then solve for k
To rid ourselved of the square roots we'll square both sides to get:
[sq root(9+ sq root 80) + sq root( 9- sq root 80)]^2 = k^2

When we use the FOIL techniques above we get: 20 = k^2
which means root20 = k
Or 2root5 = k
However I don't see 2root5 as an option
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by Brent@GMATPrepNow » Mon Aug 09, 2010 12:28 pm
emiflo wrote:sq root(9+ sq root 80) + sq root( 9- sq root 80) equals

a. 1
b. 9- 4 sq root 5
c. 18- 4 sq root 5
d. 18
e. 20

Hope the question is understandable. I'm trying to say that sq root = ^1/2. hope someone can solve this ASAP.Thanks
The other approach here would be to estimate. We can do this quickly.
Note that the root80 is very close to 9, but it must be less than 9 since root81=9
Let's say that root80 = 8.something
So, we get: root(9+ root 80) + root( 9- root 80) = root(9 + 8.something)+ root(9 - 8.something)
= root(17.something)+root(0.something)
= 4.something + 0.something
= some value around 5 (somewhere in the 4.something to 5.something range)
As long as there is only one answer in this range then we're fine.

In this question the correct answer (root20 or 2root5) isn't provided, but if the answers were sufficiently spread out, our estimation would yield only one legitiate answer.
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