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McGraw-Hill's GMAT 2013 Chapter 2 Question 10

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McGraw-Hill's GMAT 2013 Chapter 2 Question 10

by chrisontherun » Wed Apr 03, 2013 2:34 pm
If the square root of t is a real number, is the square root of t positive?
(1) t > 0
(2) t^2 > 0

Answer is apparently E (neither sufficient)

statement (2) is understandably insuff - t could be negative;
but statement (1), I think is sufficient. I quote wiki: "The principal square root function (usually just referred to as the "square root function") is a function that maps the set of non-negative real numbers onto itself."
if we know the number is positive, the square root is real then the square root is also positive.

Is this a mistake in the book? Or am I not getting something?
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by Anju@Gurome » Wed Apr 03, 2013 9:52 pm
chrisontherun wrote:If the square root of t is a real number, is the square root of t positive?
(1) t > 0
(2) t² > 0
If square root of t is a real number, t is non-negative.
Hence, t is either zero or positive.

Now, consider the following two cases,
  • square root of t = 1 ---> t = 1 > 0 ---> t² > 0
    square root of t = -1 ---> t = 1 > 0 ---> t² > 0
Both of the above examples satisfy both the statements, but square root of t is positive in the first case and negative in the second case.

The correct answer is E.

chrisontherun wrote:but statement (1), I think is sufficient. I quote wiki: "The principal square root function (usually just referred to as the "square root function") is a function that maps the set of non-negative real numbers onto itself."
if we know the number is positive, the square root is real then the square root is also positive.
There is a difference between square root of x² and √(x²).
There are two square roots of x² : one is positive and the other one is negative. They are x and -x
Among these, the positive one is the 'principal square root' of x². Which is denoted as √(x²).

By definition, √(x²) is always positive and given as |x|.
And two square roots of x² are √(x²) = x and -√(x²) = -x

The portion of Wikipedia you have quoted is actually saying : The principal square root function f(x) = √x (usually just referred to as the "square root function") is a function that maps the set of non-negative real numbers onto itself.
It is actually talking about √x which is different from square root of x, as I have explained earlier.

Hope that helps.
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by chrisontherun » Wed Apr 03, 2013 10:00 pm
Wow...this is a definition issue. Well I find that a bit underwhelming, but I do appreciate your help and your speedy reply.

Cheers,

Chris
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by GMATGuruNY » Thu Apr 04, 2013 3:54 am
chrisontherun wrote:If the square root of t is a real number, is the square root of t positive?
(1) t > 0
(2) t^2 > 0

Answer is apparently E (neither sufficient)

statement (2) is understandably insuff - t could be negative;
but statement (1), I think is sufficient. I quote wiki: "The principal square root function (usually just referred to as the "square root function") is a function that maps the set of non-negative real numbers onto itself."
if we know the number is positive, the square root is real then the square root is also positive.

Is this a mistake in the book? Or am I not getting something?
I would ignore this question.

On the GMAT, "√" means the NONNEGATIVE root only.
√4 = 2.
√x = the NONNEGATIVE ROOT of x.
√(x²) = the NONNEGATIVE ROOT of x² = |x|.

The question stem above can reasonably be interpreted as follows:
Is √t > 0?.
Since √t means the NONNEGATIVE ROOT of t, and each statement here implies that t≠0, it must be true in each case that √t > 0.

While √x means the nonnegative root of x, we must be aware of the following distinction:
If x²=4, then x=2 OR x=-2.
We must consider BOTH solutions for x.

To sum up, if x≠0:
√x implies only ONE SOLUTION (the NONNEGATIVE solution).
√9 = 3.

x² implies TWO SOLUTIONS (both the positive AND the negative solution).
x²=9 means x=3 OR x=-3.
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by Brent@GMATPrepNow » Thu Apr 04, 2013 6:01 am
To add to Mitch's post, I thought I'd post an excerpt from the Official Guide:

A square root of a number n is a number that, when squared, is equal to n. The square root of a
negative number is not a real number. Every positive number n has two square roots, one positive
and the other negative, but sqrt(n) denotes the positive number whose square is n. For example, sqrt(9) denotes 3.

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