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

## PERFECT ALGBRA

##### This topic has expert replies

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

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

--Anand--

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### GMAT/MBA Expert

- Brian@VeritasPrep
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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!

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!

Brian Galvin

GMAT Instructor

Chief Academic Officer

Veritas Prep

Looking for GMAT practice questions? Try out the Veritas Prep Question Bank. Learn More.

GMAT Instructor

Chief Academic Officer

Veritas Prep

Looking for GMAT practice questions? Try out the Veritas Prep Question Bank. Learn More.

### GMAT/MBA Expert

- Brian@VeritasPrep
- GMAT Instructor
**Posts:**1031**Joined:**03 Jul 2008**Location:**Malibu, CA**Thanked**: 716 times**Followed by:**255 members**GMAT Score:**750

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.

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.

Brian Galvin

GMAT Instructor

Chief Academic Officer

Veritas Prep

Looking for GMAT practice questions? Try out the Veritas Prep Question Bank. Learn More.

GMAT Instructor

Chief Academic Officer

Veritas Prep

Looking for GMAT practice questions? Try out the Veritas Prep Question Bank. Learn More.

### GMAT/MBA Expert

- Brent@GMATPrepNow
- GMAT Instructor
**Posts:**14908**Joined:**08 Dec 2008**Location:**Vancouver, BC**Thanked**: 5254 times**Followed by:**1263 members**GMAT Score:**770

I like the solutions I've seen so far, except there seems to be a problem.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

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

Brent Hanneson - Creator of GMATPrepNow.com

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- Brent@GMATPrepNow
- GMAT Instructor
**Posts:**14908**Joined:**08 Dec 2008**Location:**Vancouver, BC**Thanked**: 5254 times**Followed by:**1263 members**GMAT Score:**770

The other approach here would be to estimate. We can do this quickly.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

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.

Brent Hanneson - Creator of GMATPrepNow.com

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And check out these

**If you enjoy my solutions, I think you'll like my GMAT prep course**Watch these

**video reviews**of my courseAnd check out these

**free resources**