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GMAT Quant: Roots

Master these seemingly intimidating mathematical symbols!

Practice Questions

First, try these practice questions.

1) The numbers a, b, and c are all non-zero integers. Is a > 0?

Statement #1: a=b^2
Statement #2:  a=sqrt{c}

2) The numbers a, b, and c are all non-zero integers. Is a > 0?

Statement #1: a=b^3
Statement #2:  a=root{3}{c}

 

Square Roots

When you square a number, you are multiplying it by itself, e.g. 6*6 = 36.  When you take the square-root of a number, you are undoing the square, going backwards from the result of squaring to the input that was originally squared: sqrt{36} = 6.  Similarly, 8*8 = 64, so sqrt{64} = 8.  As long as all the numbers are positive, everything is straightforward.

It’s easy to find the square root of a perfect square.  All other square roots are ugly decimals.  For estimation purposes on the very hardest GMAT questions, it might be useful to memorize that sqrt{2} = 1.4 and sqrt{3} = 1.7, but without a calculator, no one is going to ask you to calculate the values of any decimal square roots bigger than that.  If something like sqrt{52} shows up, all you have to recognize is between what integers you would find that decimal.  For example,

49 < 52 < 64

therefore

sqrt{49} < sqrt{52} < sqrt{64}

therefore

7 < sqrt{52} < 8

The Symbol: Positive or Negative?

 

What is the name of this symbol?  The benighted unwashed masses will call this simply the “square root symbol”, but that’s not the full story.  The technical name is the “principal square root symbol.” Here, “principal” (in the sense of “main” or “most important”) means: take the positive root only.

This thickens the plot.  The equation x^2 = 16 has two solutions, x = +4 and x = -4, because 4^2 = 16 and (-4)^2 = 16, and the GMAT will impale you for only remembering one of those two.  At the same time, sqrt{16} has only one output: sqrt{16} = +4 only.  When you yourself undo a square by taking a square root, that’s a process that results in two possibilities, but when you see this symbol as such, printed as part of the problem, it means find the positive square root only.

Notice that we can take the square root of zero: 0^2= 0, so sqrt{0} = 0, perfectly legal.  Notice, also, we cannot take the square root of a negative — for example, sqrt{-1} — because that involves leaving the real number line.  There are branches of math that do this, but it’s well beyond the scope of the GMAT.

 

Cubes and Cube Roots

When we raise a number to the third power, 2^3, that is called “cubing” it (because if we had a cube of side = 2, then the “cube” of that number would equal the volume of the cube).  Here, 2^3 = 8.  A cube root simply undoes this process: root{3}{8} = 2.  As with a square roots, it’s easy to find the cube roots of perfect cubes, and on the GMAT you would never be expected to find an ugly decimal cube root without a calculator.

Cubes and cube roots with negatives get interesting.  While 2^3 = 8, it turns out that (-2)^3 = -8.   When you multiply two negatives you get a positive, but when you multiply three negatives, you get a negative.  More generally, when you multiply any even number of negatives you get a positive, but when you multiply any odd number of negatives, you get a negative.  Therefore, when you cube a positive, you get a positive, but if you cube a negative, you get a negative.

This means: while you can’t take the square root of a negative, you certainly can take the cube root of a negative.  Undoing the equation (-2)^3 = -8, we get root{3}{-8} = -2.  In general, the cube root of a positive will be positive, and the cube root of a negative will be negative.

It can also be a time-saver to remember the first five cubes:

1^3 = 1

2^3 = 8

3^3 = 27

4^3 = 64

5^3 = 125

You generally will not be expected to recognize cubes of larger numbers.  Knowing just these will translate handily into all sorts of related facts: for example, root{3}{125} = 5 and root{3}{-125} = -5.

If you remember just this, you will be well-prepare for whatever the GMAT asks you about roots.

 

Additional Practice Question

3) http://gmat.magoosh.com/questions/91

 

Explanations of the Practice Questions

1) All that is given in the prompt is that a, b, and c are non-zero integers.

Statement #1: the result of square anything is always positive, so whether b is negative or positive, a must be positive.  This statement, by itself, is sufficient.

Statement #2: since the square root symbol is printed as part of the problem, the output of the sqrt{c} must be positive.  We know for a fact that a must be positive.  Again, this statement, by itself, is sufficient.

Both statements are sufficient.  Answer = D.

2) Again, all that is given in the prompt is that a, b, and c are non-zero integers.

Statement #1: now, if we cube a positive, we get a positive, but if we cube a negative, we get a negative.  The numbers a & b are either both positive or both negative, but since we don’t know the sign of b, we cannot determine the sign of a.  This statement, by itself, is insufficient.

Statement #2: if we take the cube root of a positive, we will get a positive, but if we take the cube-root of a negative, we get a negative.  The numbers a & c are either both positive or both negative, but since we don’t know the sign of c, we cannot determine the sign of a.  This statement, by itself, is insufficient.

Combined Statements: If we put both statements together, we get that all three numbers, a, b, and c, have to have the same sign: either all three are positive, or all three are negative.  We have no further information that would allow us to determine which of those two is the case.  Thus, even with combined statements, we still do not have enough information to give a definitive answer to the prompt question.  Combined, the statements are still insufficient.  Answer = E

About the Author

Mike McGarry is a Content Developer for Magoosh with over 20 years of teaching experience and a BS in Physics and an MA in Religion, both from Harvard. He enjoys hitting foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Follow him on Google+!

6 Responses to GMAT Quant: Roots

  1. Kumar September 19, 2014 at 9:27 pm #

    Hi Mike,

    Since we cannot take the square root of a negative number .e.g. sqrt (-1). Here Quest 1 just says a, b & c are non-zero integers. So can’t ‘c’ be equal to ‘-1′ ? If yes, ans to ques 1 should be A.

    Or do we simply consider anything coming out of the square root always positive. I understand that same has been explained in the previous comment(s). But still it is not 100% clear to me.
    If we consider order of operation, i.e. [ (-1)^2 -> 1 -> sqrt(1) -> 1 (always positive)] , why would we ever have situation like sqrt (-1) ? And if we can have such situation, why are we not considering it in Statement 2 of Question 1.
    Can you please elaborate.

    • Mike
      Mike September 20, 2014 at 11:55 am #

      Dear Kumar,
      I’m happy to respond. :-) Statement #2 says that a = sqrt(c). In other words, it is giving us a mathematical equation as a factual, sensible, valid statement. We don’t know what a & c equal, but we know that this equation is a factual, valid mathematical statement. Well, if c = -1, then that statement wouldn’t be valid: it would be mathematical nonsense, and would depart from the real number system. (There are other branches of mathematics in which one can take the square root of a negative, but in GMAT math, it’s 100% forbidden.) This is an extremely subtle thing about math — if the GMAT gives us an equation as true, then we may not know all the values of the variables, but we know that whatever the values are, the equation must work and be sensible. If the GMAT tells me that for some numbers, P = 1/Q, then I absolutely know, in order for this to be a sensible and valid mathematical statement, that neither P nor Q could possibly equal zero. If an equation is given as true, we can derive constraints on the variables, the very constraints that guarantee that equation is valid & sensible & mathematical legal in every way. That’s why, in statement #2, we absolutely know that c must be positive, and a is the square root of a positive, which is also positive.

      In terms of GMAT math, the square root of a negative number is something outside the realm of mathematical meaning. If square root of a negative shows up in any way, know that at this point you are no longer in the realm where mathematical sensible statements are possible: this square root of a negative is something that breaks the mathematical law and is prevented from ever existing in a sound logical mathematical way. We can always automatically eliminate that even as a possibility.

      As for your other questions, my friend, I think it will help you to read the GRE blog very carefully:
      http://magoosh.com/gre/2012/positive-and-negative-square-roots-on-the-gre/
      Don’t worry about the GRE question type at the beginning of that article. Read that article very carefully, and I think it will clear up your remaining doubts.
      Mike :-)

      • Kumar September 21, 2014 at 5:26 am #

        Thanks Mike. Cheers ! :)

        • Mike
          Mike September 21, 2014 at 12:36 pm #

          Kumar,
          You are quite welcome. :-) Best of luck to you.
          Mike :-)

  2. Sunny April 14, 2014 at 10:46 pm #

    Hi,

    I was wondering, for question 1, couldn’t we make c = -1^2?

    • Mike
      Mike April 15, 2014 at 10:32 am #

      Dear Sunny,
      I’m happy to respond. First of all, with all due respect, I think you need some work with parenthesis, because -1^2 means -(1^2), but I suspect what you meant was (-1)^2. Parentheses are not garnish: they are vitally pieces of mathematical equipment. See:
      http://magoosh.com/gmat/2013/gmat-quant-mathematical-grouping-symbols/
      The sloppy unsophisticated way of thinking about a square root is that it “undoes the square.” That naive view is loaded with fallacies that the GMAT loves to exploit, such as, I am sorry to say, the very fallacy into which you fell when you posed this question. I am being very direct here because I really want to understand the nature of this mistake so you avoid it in the future.
      In fact, the Order of Operations is always in effect; we cannot escape the strict authority of the Order of Operations. We start with (-1), then we square it, which of course is c = +1. Only after that do we take a square root of c = +1, which of course is +1.
      Does all this make sense?
      Mike :-)


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