## Practice problems

First of all, consider these two similar, but not identical, Quantitative Comparison questions.

All I will say right now is: despite apparent similarities, those two questions have two completely different answers. The distinction between them is the subject of this article.

## Do I include the negative root?

Often, students are confused about this question. For example, in the questions above, we know +4 is a square root of 16, but isn’t -4 as well? Do we include it as part of Column B or not? Does it matter how the question is framed? All of these questions are resolved by understanding the following two cases.

## Case I

In case one, the test-maker, in writing the question, uses this symbol. This symbol appears printed on the page as part-and-parcel of the question itself.

What is this symbol? Well, the benighted masses call this simply a “square-root” symbol, but the proper name is the “principal square root” symbol. Here, “principal” (in the sense of “main” or “most important”) means: you take one and only one root, the most important one —i.e. the positive root only. That is the deep meaning of this symbol.

Thus, in this case, cases in which the test maker, by the sheer process of writing the question itself, has written down this symbol, and this symbol appears as part-and-parcel of the question itself, then you NEVER consider the negative root, and ONLY take the positive root.

## Case II

In this case, that special symbol does **not** appear as part of the problem. What does appear is, possibly, a variable squared, or some other combination of algebra that leads to a variable squared, and you yourself, in your process of solving the problem, have to take the square-root of something in order to solve it. The act of “square rooting” is not initiated by the test maker, in the very act of writing the question, but is left to you, the problem solver, to initiate.

In this case, 100% of the time, you ALWAYS have to consider **both** the positive and negative square roots.

## Summary

If you master that distinction, you will always understand when to consider both positive and negative roots, and when you need only consider the positive root. You may want to go back to those two QCs at the beginning and think them through again before reading the solutions below.

## Practice problem solutions

1) Here, the principal square root symbol appears printed as part of the problem itself. We are in Case I. Of course, that symbol implies: take the positive square root only. So Column B can only equal +4. Of course, that’s always bigger than 3, so Answer = **B**.

2) Here, there’s no square root symbol printed as part of the problem itself. We are in Case II. For any square roots we take as part of our solution, we are liable to account for both the positive and negative roots.

Sure enough, the very first thing we encounter in the prompt is a variable squared, and when we solve for x, we have to account for both roots: x = ±4. The variable x could equal well have either one of those values.

Now, when we proceed to the QC, we see that the different values of x would give different answers. If x = +4, then column B is greater, but if x = -4, then column A is greater. Different values lead to different conclusions, and this situation means we don’t have enough information to establish a definitive relationship. Answer = **D**

Hi,

I have a quick question regarding QC which actually deals with extraneous roots.

What if we have two comparions , A and B .

A = Has an equation which has no solution.

B = Has an equation which has one solution.

Does this mean B > A ?

For example:

A) 2 + âˆš (4-3x) = x

B) âˆš (x+3) = x-3

Since A does not have any solution, and B has one.

Does this mean B > A ?

It would be really helpful if you could help with this.

Kind Regards,

Wali

Dear Wali,

My friend, you have to realize that GRE math questions are extremely precise in their formulation. Notice that what you have produced, as is, is

a legitimate QC question. One of the columns in a QC question isnotan equation to solve. The prompt may give an equation, and then in the column itself a variable from the equation might appear. Comparing one equation to another equation makes no sense at all, and is not GRE-like at all. You may find algebraicneverexpressionsin the columns, but you will never find twoequationsin the columns.Let’s reformulate your question in a couple of different ways.

Reformulation #1Prompt:

Equation #1: 2 + sqrt(4 – 3x) = x

Equation #2: sqrt(y + 3) = y – 3

Column A: number of possible values of x that satisfy equation #1

Column B: number of possible values of y that satisfy equation #2

OK, the GRE would never ask something like this in a million years, but at least it’s now in legitimate QC form. Numerical value of Column A would be zero, and the numerical value of Column B would be 1, so the answer would be

B.Reformulation #2Prompt:

2 + sqrt(4 – 3x) = x

sqrt(y + 3) = y – 3

Column A: x

Column B: y

The GRE would never ask this in a billion years, but again, this is at least in something vaguely resembling QC format now. In Column B, we have a numerical value of 6. In Column A, we have something that has absolutely no mathematical meaning. We are being ask to compare six to meaninglessness! Which is bigger, six or meaninglessness? This is inherently a nonsensical comparison (one reason the test would never give it to us), and within the laws of mathematics, there is no sensible way to assign definitive relationship. Thus,

Dwould be the only possible answer.It’s very important to distinguish a real value of zero, x = 0, from a situation in which x simply has no value. The value x = 0 is a real number that lives on the real number line. If an equation has no solution, then there is no value of x anywhere on the number line that satisfies it: in this case, the “value of x” is a meaningless quantity, and pure meaninglessness is extremely different from the fixed numerical value of zero.

Does all this make sense?

Mike

Does anyone else think it’s totally outrageous for the test makers to use this cheap trick to hiccup test takers who haven’t learned this rather absurdly specific definition? It’s a general knowledge test…not a test of the nuances of mathematical terms. Totally inappropriate imo.

Hi Mike,

If (x/y)< (1/2): can we rewrite this as: 2x<y? In reference to this question-http://gmat.magoosh.com/practices/8337794/q/970?prompt_id=970

Dear Arefin,

I’m happy to respond. With inequalities, we are ALWAYS, 100% of the time, allowed to multiply both sides by a POSITIVE number. This means we are absolutely allowed to multiply by 2, changing

(x/y) > (1/2)

to

(2x/y) > 1

Now, the sticky question is: can we also multiply both sides by y? Well, the pertinent question is: is y positive? Hmm. We simply don’t know, because y is a variable: it could be positive or negative. We would break the mathematical law if we multiplied both sides by a negative and didn’t reverse the inequality. Because we don’t know the sign of y, we don’t know whether we would be breaking the mathematical law by multiplying both sides by it. Therefore, multiplying both sides by y is forbidden.

In that question, statement #1 is insufficient, because we are NOT allowed to cross multiply with a variable about which we know nothing.

Statement #2 tells us both variables are positive, so when we combine the statements, we now have the guarantee that y is positive, which allows us to cross-multiply.

In terms of numbers if

(x/y) > (1/2)

then it could be that x = 4 and y = 7, so then it definitely is true that 2x > y.

BUT, it also could be true that x = -4 and y = -7, in which case it’s true that 2x = -8, and 2x < y, because (-8) < (-7).

Does all this make sense?

Mike

Thanks Mike…..Very helpful. I get the point.

ERUDITE TEACHER

Dear Jatinder,

Thank you for your kind compliment! Best of luck to you!

Mike

Dear Mike,

In your blog post, you state:

“First of all, consider these two similar, but not identical, QCs”.

The embedded link is to the following website or location: http://magoosh.com/gre/category/math-question-types/qc-quantitative-comparison/

Unfortunately, this is just the main page where a series of math questions, strategies etc. regarding QCs have been discussed and posted. May you kindly please provide the actual link to the 2 QC questions on positive and negative square roots?

Thanks in advance!

Kindest,

Samy

Dear Samy,

I’m happy to respond. That link should not have been there: I just removed it. The two “

similar but not identical” QCs to which I am referring are those two at the top of this article. I am not referring to anything that is not already on this page.Does this make sense?

Mike

Hi Mike

It doesn’t make any sense to take only positive values. Is it anywhere mentioned by ETS that we have to perform such an operation when we encounter the square root sign?

Going by the conventional rules, any number treated with a square root sign will give both positive and negative values.

I’m not judging anyone here but it seems a little bit difficult to digest this fact.

Regards

Aman Sachan

hi mike ,

thanks for the info . it has helped a lot

Dear Chinmay,

You’re quite welcome. I’m glad you found it helpful. Best of luck to you.

Mike

Hi,

If the question is 4 raised to 1/2 then we have to consider both positive and negative Or not??

[when we see square root SYMBOL then only positive???]

Thanks

Dear Manoj,

Whether the problem gives you 4 under the radical or 4 to the power of (1/2), either way, the problem-writer is the “initiator” of the square-rooting process, and therefore you only consider the positive root. Does this make sense?

Mike

Thanks for the reply Mike.

I came across a problem to compare

(1/2) ^ 2 (1/16 )^1/2

and answer was (d) and the explanation included the negative root. Since I had already visited this page i chose A [and i was invigorated when it said i was wrong, after putting all the conscious effort ].

This happened just before 5 hours of my GRE test

[ I would have chose D if i had witnessed (^1/2) this on the test ]

I told myself in GRE “what you see is what you get”. so i used my GRE mind

and concluded that case 1 on top of this page talks ONLY about the symbol.

I had convinced myself that if i SEE the ROOT symbol I will take only positive.

On the contrary if i see (^1/2) i will consider both.

Fortunately i did not witness any question related to roots using these symbols in comparison.

I understand that its the problem writer who initiates the process. The dilemma started when i saw that problem.

Please suggest a final call on this issue.

[According to you its final and binding that if its a square root unless otherwise mentioned + or – take positive right? and that would mean that their explanation is wrong ]

I hope this helps the test takers.

Many Thanks

Mike

Dear Manoj,

The GRE OG doesn’t even discuss fractional exponents. This is a very rare topic on the GRE — you might be able to take 10 GREs and not see fractional exponents once. That’s just to put the relative importance in context.

Every standard high school math book on the planet defines (a)^(1/2) exactly the same way as “a” under a radical — both mean positive root only. ETS absolutely has to follow that convention, because they are not out to be cheap and to trick people with bizarre alternative technicalities of convention.

ETS is not cheap and tricky, but some of the folks who write GRE practice questions are cheap and tricky, and it sound like this question is in this vein. Unfortunately, there’s a great deal of low quality material out there that can be confusing.

Finally, in this question,

Column A: (1/2)^2

Column B: (1/16)^(1/2)

I would say the correct answer would have to be (C), because both columns would equal +1/4.

Does all this make sense?

Mike

Oh yes,

with such a good and elaborate explaination it definitely made sense.

You are right GRE is definitely not cheap and vague.

I loved taking the exam.

Thanks

Mike

You are quite welcome. Best of luck to you.

Mike

This is extremely important and I saw no mention of this in the Magoosh lessons. Thanks for clearing that up!!

Shane,

You are more than welcome. Best of luck to you.

Mike

Mike for PRESIDENT!!!

Dear Tamaddun,

Thank you for your kind words, but I’m not really interested in holding even a local political office, let alone Mr. Obama’s job! I would much rather help people prepare for the GMAT & GRE. That’s much more fun! Best of luck to you!

Mike

Wow !!!! You are a ” PRINCIPAL” !!!! Great explanation.

Dear Sandeep,

Thank you very much for your kind words. Best of luck to you, my friend.

Mike

what if it says 16^(1/2) ? is this ambiguous or not

Dear tt,

I don’t believe the GRE expects you to know this, but by convention, any fractional power of a positive number has a positive-only output.

Mike

what if he didnt mention the symbol and just wrote, square root of the number 25,answer is both + and – 5?

Dear Siddharth,

First of all, I am a little unclear on the identify of the “he” in your question — do you mean the GRE Test maker? (The GRE is assembled by many people.)

I have never seen the GRE test present all the mathematical information about roots in VERBAL form and then expect you to make valid mathematical deductions from it.

Mike

This is the exact thing I was looking for since several months. Thanks a lot. Enlighten. ðŸ˜€

Dear Amey,

Thank you for your kind words. Best of luck to you.

Mike

I was benighted of this trick !! Thnxx a lot Mike

You are quite welcome, my friend.

Mike

Don’t you guys(fellow Magoosh readers!) love it when the AWESOME Magoosh “magi” help us improve our vocabulary in a super tricky math concept article?

I just learned “benighted” thanks to you Mike, Cheers!

Thanks for making the distinction (I would have missed that). I just became a member today and I already find the lessons and explanations invaluable. I feel like I’m stealing from you guys (don’t get any ideas). All kidding aside, thanks for doing what you do because I was going the Kaplan route, but the Kaplan route is not for me.

Congratulations on joining Magoosh: I think you will see tremendous benefit in your score. Thank you very much for your kind words.

Mike

Thanks Mike ! I was always confused with such questions. This post makes things a lot clearer !

You are quite welcome.

Mike

Wow! Thanks for these blog posts. The more I read, the more I am amazed at my own ignorance. I also start to feel a bit lost in the sea of things that I don’t know.

Have faith, my friend. You can master this stuff! Thank you for your kind words.

Mike

Thanks Mike — Really helpful.

You are very welcome.

Mike

Hi Mike, it’s great that you pointed out that the radical sign âˆš refers to the principal (positive) square root only. (many people use it without realizing that)

In the quadratic equation, Â± is used in front of âˆš (b2-4AC) so that the negative root is included in the equation…..and not just the positive root.

source: square root calculator

Anthony:

Exactly: the âˆš sign means “positive only”, so in any context in which both the Â± roots are required (as in the Quadratic Formula), we need to add the Â± sign in front of the âˆš . What you have shared is 100% correct. Thank you.

Mike

Hi Mike, Many thanks for pointing out this important difference . It seemed innocuous , and most people (including me) would have failed to realize this fine distinction.

Thanks to you, now I’ll be on a lookout

Regards,

Anupam

You are quite welcome. Thank you for your kind words.

Mike