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help in numper prop. important yet confusing square root

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by jaymin » Sun Jan 25, 2009 10:01 pm
Hi, I need help in understanding the following two:

1.Does the square root of a number have 2 values or 1 value. If a DS question is asked as follows:
What is the value of 3x+2?
a. square root 4 = x.

Then, is the first statement sufficient or is x= +,- 2.? I interpret it this way- root 4=x => 4=x square => x= +,-2. I read the og11 math review definition but could not get a clear understanding, since it mentions that root 4=x has 2 values viz. + and - 2, but the positive root of 4 has only one value +2. So, with this definition, will statement (1) in above Question be suffcient?

2. Also, Is 0 considered a perfect square? OR does the perfect square series begin with 1,2,4,9....

Thanx
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by gaggleofgirls » Mon Jan 26, 2009 10:06 am
A square root has 2 values a positive and a negative becuase a pos*pos = pos and a neg*neg = pos Without knowing more than just sqrtX you can't say it is a pos number (and this seems to come up in a lot of the gmat problems).

so, if choice a in your example was |sqrt4|, then you would know it was the positive sqrt 4. But without that, there are 2 answers and a is not sufficient.

For a DS question, you might have

What is 3x=2?

1) x = sqrt 4
2) x > 0

either alone are insufficient, but together they are sufficient, so C.

And from what I have seen on this forum, that would be considered an easy-mid problem, not a hard one.


Also, x^2 will always be pos, but x^3 may be negative (is x is neg). To expand, to an even power will always be positive, x to an odd power will be the sign of X. This also seems to come up a lot.

Both of these seem to be very important to data sufficiency problems.

As for 0 being a perfect square, I haven't seen that yet on the gmat in my studying. I believe it is a debate in math circles.

-Carrie
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by jaymin » Mon Jan 26, 2009 11:21 am
Thanx Carrie.

I agree with you, since root 4=x, now squaring both sides => 4= x^2 and thus x is + or - 2.

But, in the Manhattan gmat number properties guide, it is mentioned that the square root of a number is positive. Also, in the OG 11 math review, it is mentioned that square root stands for the positive square root only. In such a case, on the gmat, isn't statement (a) sufficient?

can we have more light on this?
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by jaymin » Mon Jan 26, 2009 11:22 am
Thanx Carrie.

I agree with you, since root 4=x, now squaring both sides => 4= x^2 and thus x is + or - 2.

But, in the Manhattan gmat number properties guide, it is mentioned that the square root of a number is positive. Also, in the OG 11 math review, it is mentioned that square root stands for the positive square root only. In such a case, on the gmat, isn't statement (a) sufficient?

can we have more light on this?
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by jaymin » Mon Jan 26, 2009 11:22 am
Thanx Carrie.

I agree with you, since root 4=x, now squaring both sides => 4= x^2 and thus x is + or - 2.

But, in the Manhattan gmat number properties guide, it is mentioned that the square root of a number is positive. Also, in the OG 11 math review, it is mentioned that square root stands for the positive square root only. In such a case, on the gmat, isn't statement (a) sufficient?

can we have more light on this?
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by gaggleofgirls » Mon Jan 26, 2009 1:35 pm
I haven't used either of those guides so someone else will have to chime in.

-Carrie
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by Ian Stewart » Mon Jan 26, 2009 2:26 pm
The distinction that's important here is the use of the square root symbol. If you see the square root symbol over a positive number x, that means 'the positive square root of x'. So if you see the square root symbol over the number 9, that is equal to 3, and only 3. That is how the root symbol is defined. Still, 9 does have two square roots, 3 and -3, because 3^2 = 9, and (-3)^2 = 9. So, if I say in words 'x is a square root of 9', x could have two values, and if you see the equation x^2 = 9, then x could be 3 or -3, but if you see the square root symbol with a 9 underneath, that is only equal to 3, and not -3.

The above is precisely what the OG says, incidentally, when it defines roots on page 114, in the middle of the page.
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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