What is the value of sqrt (x)? EXPERTS PLEASE HELP !!

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rahulg83: Legendary Member; Posts: 575; Joined: Tue Nov 04, 2008 2:58 amFollowed by: 4 members; Location: India; Thanked: 18 times; GMAT Score:710

by rahulg83 » Sun Jun 28, 2009 6:32 pm

Answer should be A if sqrt(16)=4, statement 1 is sufficient.

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cameronwu: Senior | Next Rank: 100 Posts; Posts: 34; Joined: Wed Jun 17, 2009 6:15 am; Thanked: 4 times

by cameronwu » Wed Jul 01, 2009 7:21 am

Domnu wrote:Btw, I don't think the answer is C... the answer should be A.

If sqrt(x) is an integer, then x has to be an integer. So, x = 12, 13, 14, 15, 16 only. But 16 is the only perfect square, and sqrt(16) = 4 (not -4). So, the answer is A.

If you know what imaginary numbers are, this is where they come in. But if you're not sure what this is, then never mind.

I certainly know what imginary numbers are, but I'd like to know how they come into effect with this problem?

Sqrt(-16) is an imaginary number... -4 squared is positive 16. Where is the imaginary concept in the problem?

Despite what everybody says about "GMAT Land" and assuming that only a positive root exists - I Call BS. Does it disclaim that on the test? It logically doesn't make sense for them to grade questions on that basis:
1) It's contrary to general rules of mathematics
2) If it's not disclaimed, then it requires "insider info"
3) Most GMAT mathematics problems, like the word problems, will specify if x is a positive integer only. Why would the case be different for data sufficiency?
4) How does the song to quadratic equation go? Oh yeah, PLUS OR MINUS the square root of b^2-4ac all over 2a... Why would the laws of square roots operate differently just because it's not in the context of an algebraic equation?

Based on the above reasons, when considering the original problem, (i) is insufficient because it can be +/-4. (ii) is insufficient because it can be 3 or 4. When taken together, it has to be sufficient. C is your answer.

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jakesing: Junior | Next Rank: 30 Posts; Posts: 21; Joined: Fri Jun 12, 2009 2:10 pm; Thanked: 1 times

by jakesing » Wed Jul 01, 2009 8:17 am

Domnu wrote: Sqrt(-16) is an imaginary number... -4 squared is positive 16. Where is the imaginary concept in the problem?

Despite what everybody says about "GMAT Land" and assuming that only a positive root exists - I Call BS.

I am with you here. I think there is some really bad, misleading information going on in this thread. Look at √x, for any x. On the GMAT, x will definitely be positive. But the result, √x, will be either the positive or the negative square root. It is absolutely incorrect to say that √9 only equals 3 on the GMAT. (-3) × (-3)=9, even on the GMAT. The answer to the original question, as you write, is C.

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madhur_ahuja: Master | Next Rank: 500 Posts; Posts: 435; Joined: Sat May 02, 2009 3:55 am; Thanked: 17 times

by madhur_ahuja » Wed Jul 01, 2009 10:12 am

It is absolutely incorrect to say that √9 only equals 3 on the GMAT

I agree with C, but above statement is wrong. Can anyone confirm?
In my opinion, square root always means positive square root.

Clarification:
y^2= 9
y = +/- sqrt(9) = +/- 3

But,
sqrt(9) = 3 only, and not -3

In above question. If we choose x as 16, then
x=16
However, we are given that sqrt(x) is an integer.
Now, sqrt(16) and -sqrt(16) are both integers.

Statement II is needed to consider positive one. And hence , answer is C

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

by Ian Stewart » Wed Jul 01, 2009 11:08 am

The square root symbol is defined as a function: it only produces one answer. By definition, the square root symbol gives the *non-negative* square root of whatever's underneath the root. So if you see the square root symbol over the number 9, the answer is most certainly 3, and most certainly *not* -3. This is true in GMAT mathematics, and true in every other branch of mathematics - there are no special mathematical rules or definitions specific to the GMAT (except, perhaps, that we ignore complex numbers on the GMAT).

All of that said, 9 does have two square roots: 3 and -3. It's just that when you see the square root symbol over the number 9, we ignore the negative answer, because of how the symbol is defined.

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cameronwu: Senior | Next Rank: 100 Posts; Posts: 34; Joined: Wed Jun 17, 2009 6:15 am; Thanked: 4 times

by cameronwu » Wed Jul 01, 2009 11:31 am

Ian Stewart wrote:The square root symbol is defined as a function: it only produces one answer. By definition, the square root symbol gives the *non-negative* square root of whatever's underneath the root. So if you see the square root symbol over the number 9, the answer is most certainly 3, and most certainly *not* -3. This is true in GMAT mathematics, and true in every other branch of mathematics - there are no special mathematical rules or definitions specific to the GMAT (except, perhaps, that we ignore complex numbers on the GMAT).

All of that said, 9 does have two square roots: 3 and -3. It's just that when you see the square root symbol over the number 9, we ignore the negative answer, because of how the symbol is defined.

You're suggesting the answer is (A), correct?

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by Ian Stewart » Wed Jul 01, 2009 12:00 pm

cameronwu wrote: You're suggesting the answer is (A), correct?

I hadn't even looked at the original question - I was only replying to your and jakesing's posts above - but yes, it's certainly A.

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jakesing: Junior | Next Rank: 30 Posts; Posts: 21; Joined: Fri Jun 12, 2009 2:10 pm; Thanked: 1 times

by jakesing » Wed Jul 01, 2009 3:09 pm

Ok So I get that that is the convention, but can someone explain to me how the following isn't a contradiction?

In the example given a few posts above, it says if y^2=9, y= +/-3. But if y=√9, then y=3. Maybe it's just me, but I'm pretty certain those two statements are equal. The way you solve the first one is by taking the square root of both sides, resulting in y=√9. I'm really perplexed right now.

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by Ian Stewart » Wed Jul 01, 2009 5:04 pm

jakesing wrote:Ok So I get that that is the convention, but can someone explain to me how the following isn't a contradiction?

In the example given a few posts above, it says if y^2=9, y= +/-3. But if y=√9, then y=3. Maybe it's just me, but I'm pretty certain those two statements are equal. The way you solve the first one is by taking the square root of both sides, resulting in y=√9. I'm really perplexed right now.

You can't solve the equation y^2 = 9 as you're doing. You're assuming that √y^2 is equal to y, and it is not necessarily equal to y:

*If y is positive, then √y^2 = y
*If y is negative, then √y^2 = -y
*No matter what y is, √y^2 = |y|

You'll find a couple of questions in GMATPrep that test if you understand the above. Notice that there's no reason why y can't be negative here; since we have y^2 under the root, we still have a positive quantity under the root. And if y is negative, then √y^2 cannot be equal to y; the √ of something can never be negative, so can't possibly be y.

It's surely easier to see how this works by looking at a numerical example: take, say, x = -3. Then √x^2 is certainly not equal to x; that is, √(-3)^2 is not equal to -3. Instead it's equal to √9 = 3, which is the same thing as -(-3), or -x. So, when x is negative, √x^2 = -x (which is the same thing as |x|, or the 'positive equivalent' of x).

So if you want to solve the equation y^2 = 9 by applying a √ to both sides, you need to consider two cases: y is positive, and y is negative. You'll then get the two different solutions for y. Many people learn a shortcut to this process; they learn to solve an equation like:

z^2 = 7

by taking the positive and negative roots on the right side:

z = √7 or z = -√7.

That's a perfectly good way to solve.

That said, the equation y^2 = 9 is a quadratic equation; the conventional way to solve any such equation is not to apply √ to both sides; instead it is to get zero on one side and factor:

x^2 - 9 = 0
(x + 3)(x - 3) = 0
x = -3 or x = 3

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sanjana: Master | Next Rank: 500 Posts; Posts: 182; Joined: Sun Aug 02, 2009 7:19 pm; Thanked: 18 times; GMAT Score:680

by sanjana » Fri Oct 23, 2009 1:53 am

Why isnt the answer D?

1 is sufficient as pointed out.

For statement2,it says 2<sqrtx<5
The question says sqrt(x) is an integer

Cant we say from this that sqrt x = 4,hence X=16 and
2<4<5?

Please clarify.

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pmFollowed by: 614 members; Location: Toronto; Thanked: 1710 times; GMAT Score:800

by Stuart@KaplanGMAT » Fri Oct 23, 2009 9:45 am

sanjana wrote:Why isnt the answer D?

1 is sufficient as pointed out.

For statement2,it says 2<sqrtx<5
The question says sqrt(x) is an integer

Cant we say from this that sqrt x = 4,hence X=16 and
2<4<5?

Please clarify.

Why couldn't sqrt(x) = 3 and still fit the rule? 3 is an integer between 2 and 5 as well.

Stuart Kovinsky | Kaplan GMAT Faculty | Toronto

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

by Ian Stewart » Fri Oct 23, 2009 4:57 pm

sanjana wrote:Why isnt the answer D?

1 is sufficient as pointed out.

For statement2,it says 2<sqrtx<5
The question says sqrt(x) is an integer

Cant we say from this that sqrt x = 4,hence X=16 and
2<4<5?

Please clarify.

sqrt(x) could also be 3, in which case x is 9. So Statement 2 is not sufficient.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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abz: Newbie | Next Rank: 10 Posts; Posts: 1; Joined: Fri Oct 30, 2009 3:03 pm

by abz » Fri Oct 30, 2009 3:34 pm

Stuart Kovinsky wrote:
You ask a great question, because it's essential to understand the difference, especially in data sufficiency.

Let's make up a DS question to illustrate the difference:

Q: What's the value of x?

(1) x = âˆš9

(2) x^2 = 9

Step 1 of the Kaplan Method for DS: focus on the question stem.

We're asked for the value of x; not much to think about here. We need 1 and only 1 value for a statement to be sufficient and, as of right now, we have no restrictions on x at all.

Step 2 of the Kaplan Method for DS: consider each statement by itself, in conjunction with the stem.

(1) we know that the "âˆš" sign means "postive root of". 9 only has one positive root: 3. Therefore, x must be 3... sufficient.

(2) we know that if x^2 = 9, x could be +/- 3... 2 possible values, therefore insufficient.

As an algebraic aside, here's the "official" way to solve this equation:

x^2 = 9
x^2 - 9 = 0
(x + 3)(x - 3) = 0
x = -3 or +3

(1) is sufficient and (2) is insufficient: no need to combine, select (A).

Lemme get this
(-x)^2= x^2
-by commutative law (-x)^2=(-1)(-1)x^2
-and by multiplicative inverse (-1)(-1)=1
sqrt both sides
sqrt((-x)^2)=sqrt(x^2)
sqrt(x^2)=x by the standard defn

x=sqrt(9)=3 and not -3 because
if x=-3 then (-3)^2=9 but as we have seen before (-3)^2 is in fact (3)^2 therefore x=3 which implies the positive root is sufficient.

Started writing this post opposing the standard defn with this false argument (see below) then changed my mind;
False arguement;
(-3)^2=9 which is true
sqrt((-3)^2)=sqrt(9)
(-3)^(2x1/2)=9^(1//2) by standard defn sqrt(9)=3
(-3)^1=3
-3=3 came to a contradiction
the bit i am unsure about (where falseness may arise) is commutation regarding multiplying indices.

Man's confused

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Testluv: GMAT Instructor; Posts: 1302; Joined: Mon Oct 19, 2009 2:13 pmFollowed by: 164 members; Location: Toronto; Thanked: 539 times; GMAT Score:800

by Testluv » Fri Oct 30, 2009 5:44 pm

x^2 = 9
means x = pos sqrt 9 OR
x = (-1) multiplied by pos sqrt 9

sqrt 9
means pos sqrt 9 = 3 (no multiplication by negative one)

Can't take square root of negative number on GMAT (involves imaginary numbers, a concept not tested on GMAT).

It is important to know this difference on the GMAT but understanding theoretical distinction is outside the GMAT's scope because, again, it involves imaginary numbers.

...The end!

Kaplan Teacher in Toronto

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Gmatter2.0: Junior | Next Rank: 30 Posts; Posts: 29; Joined: Fri Oct 30, 2009 10:45 pm; Thanked: 1 times; GMAT Score:580

by Gmatter2.0 » Sat Oct 31, 2009 9:43 pm

Stuart Kovinsky wrote:
sanjana wrote:Why isnt the answer D?

1 is sufficient as pointed out.

For statement2,it says 2<sqrtx<5
The question says sqrt(x) is an integer

Cant we say from this that sqrt x = 4,hence X=16 and
2<4<5?

Please clarify.
Why couldn't sqrt(x) = 3 and still fit the rule? 3 is an integer between 2 and 5 as well.

Please someone make it clear, I always thought once Statement 1 is correct and Statement 2 by itself is wrong eventhough the combination of 1 & 2 can give a qualified result still the Answer is A

Am I correct....

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