GRE high-scorers might not be smarter than everyone else, but they do think about the test differently. One key difference is in how high-scorers do algebra. They make far fewer algebraic mistakes, because, either consciously or subconsciously, they use mathematical rules to check their work as they simplify. Here’s how to develop that habit yourself. Read more

Imagine that you asked a friend of yours what she got on the Quant section of the GRE. Instead of answering you directly, she said “let’s just say that 4 times my score is a multiple of 44, and 3 times my score is a multiple of 45.”

Could you tell what score she got? If not… you may need to work on your GRE translation skills!  Read more

Have you ever gotten a GRE question wrong because you thought you were supposed to take a square root and get two different numbers but the answer key said only the positive root counted? Alternatively, have you ever gotten one wrong because you took the square root and wrote down just the positive root but the answer key said that, this time, both the positive and the negative root counted? What’s going on here?

There are a couple of rules we need to keep straight in terms of how standardized tests (including the GRE) deal with square roots. The Official Guide does detail these rules, but enough students have found the explanation confusing – and have complained to us about it – that we decided to write an article to clear everything up.

Doesn’t the OG say that we’re only supposed to take the positive root?

Sometimes this is true – but not always. This is where the confusion arises. Here’s a quote from the OG 2^nd edition, page 212:

“All positive numbers have two square roots, one positive and one negative.”

Hmm. Okay, so that makes it seem like we always should take two roots, not just the positive one. Later in the same paragraph, though, the book says:

“The symbol √n is used to denote the nonnegative square root of the nonnegative number n.”

Translation: when there’s a square root symbol given with an actual number underneath it – not a variable – then we should take only the positive root. This is confusing because, although they’re not talking about variables, they use the letter n in the example. In this instance, even though they use the letter n, they define n as a “nonnegative number” – that is, they have already removed the possibility that n could be negative, so n is not really a variable.

If I ask you for the value of √9, then the answer is 3, but not -3. That leads us to our first rule.

Rule #1: √9 = 3 only, not -3

If the problem gives you an actual number below that square root symbol, then take only the positive root.

Note that there are no variables in that rule. Let’s insert one: √9 = x. What is x? In this case, x = 3, because whenever we take the square root of an actual number, we take only the positive root; the rule doesn’t change.

Okay, what if I change the problem to this: √x = 3. Now what is x? In this case, x = 9, but not -9. How do we know? Try plugging the actual number back into the problem. √9 does equal 3. What does √-9 equal? Nothing – we’re not allowed to have negative signs underneath square root signs, so √-9 doesn’t work.

Just as an aside, if the test did want us to take the negative root of some positive number under a square root sign, they’d give us this: -√9. First, we’d take the square root of 9 to get 3 and then that negative sign would still be hanging out there. Voilà! We have -3.

What else does the OG say?

Here’s the second source of confusion on this topic in the OG. On the same page of the book (212), right after the quotes that I gave up above, we have a table showing various rules and examples, and these rules seem to support the idea that we should always take the positive root and only the positive root. Note something very important though: the table is introduced with the text “where a > 0 and b > 0.” In other words, everything in the table is only true when we already know that the numbers are positive! In that case, of course we only want to take the positive values!

What if we don’t already know that the numbers in question are positive? That brings us to our second and third rules.

Rule #2: x² = 9 means x = 3, x = -3

How are things different in this example? We no longer have a square root sign – here, we’re dealing with an exponent. If we square the number 3, we get 9. If we square the number -3, we also get 9. Therefore, both numbers are possible values for x, because both make the equation true.

Mathematically, we would say that x = 3 or x = -3. If you’re doing a Quantitative Comparison problem, think of it this way: either one is a possible value for x, so both have to be considered possible values when comparing Quantity A to Quantity B.

Rule #3: √(x)² = 3 means x = 3, x = -3

Okay, we’re back to our square root sign, but we also have an exponent this time! Now what? Do we take only the positive root, because we have a square root sign? Or do we take both positive and negative roots, because we have an exponent?

First, solve for the value of x: square both sides of √(x)² = 3 to get x² = 9. Take the square root to get x = 3, x = -3 (as in our rule #2).

If you’re not sure that rule #2 (take both roots) should apply, try plugging the two numbers into the given equation, √x² = 3, and see whether they make the equation true. If we plug 3 into the equation √x² = 3, we get: √(3)² = 3. Is this true? Yes: √(3)² = √9 and that does indeed equal 3.

Now, try plugging -3 into the equation: √(-3)²= 3. We have a negative under the square root sign, but we also have parentheses with an exponent. Follow the order of operations: square the number first to get √9. No more negative number under the exponent! Finishing off the problem, we get √9 and once again that does equal 3, so -3 is also a possible value for x. The variable x could equal 3 or -3.

How am I going to remember all that?

Notice something: the first example has either a real number or a plain variable (no exponent) under the square root sign. In both circumstances, we solve only for the positive value of the root, not the negative one.

The second and third examples both include an exponent. Our second rule doesn’t include any square root symbol at all – if we have only exponents, no roots at all, then we can have both positive and negative roots. Our third rule does have a square root symbol, but it also has an exponent. In cases like this, we have to check the math just as we did in the above example. First, we solve for both solutions and then we plug both back into the original equation. Any answer that “works,” or gives us a “true” equation, is a valid possible solution.

Takeaways for Square Roots:

(1) If there is an actual number shown under a square root sign, then take only the positive root.

(2) If, on the other hand, there are variables and exponents involved, be careful. If you have only exponents and no square root sign, then take both roots. If you have both an exponent and a square root sign, you’ll have to do the math to see, but there’s still a good chance that both the positive and negative roots will be valid.

(3) If you’re not sure whether to include the negative root, try plugging it back into the original to see whether it produces a “true” answer (such as √(-3)² = 3) or an “invalid” situation (such as √-9, which doesn’t equal any real number).

* The text excerpted above from The Official Guide to the GRE 2nd Edition is copyright ETS. The short excerpts are quoted under fair-use statutes for scholarly or journalistic work; use of these excerpts does not imply endorsement of this article by ETS.

I know that for people who’ve been away from math for a while, the GRE requires a lot of refreshment on topics and skills. Even for those of us who’ve been around math all along, there may be topics we haven’t seen since high school on the exam.

Most of getting good at GRE math is practicing your skills, learning to recognize clues and patterns on the exam, and knowing what material is being tested and how it is tested. One key step is knowing the definition of math words, because those definitions often come with important restrictions.

For example, when a question starts by specifying that x is an integer, that restriction will probably be a key to the problem. There is an infinite amount of numbers that are not integers, including fractions and radicals. It’s also important to remember that integers don’t have to be positive – there are negative integers, and zero is in integer as well.

My suggestion is that you clarify the definitions, but not simply memorize them. Let’s say that I realize knowing the definition of “integer” is important, so I decide to make a flashcard that says “integer” on one side and “a member of the set of whole numbers” on the other.

Great. That’s true, and if the test were going to ask me to define the word “integer”, that would be a great thing to know. But remember: for the most part, the quant section of the GRE is a skill test, not a knowledge test. It tests your ability to notice patterns and details, perform math tasks, plan an efficient road to a solution, and reason with numbers. So the definition of “integer” that I want to know is something that will help me.

I am not the biggest fan of flashcards for the quant portion of the GRE, but if I were going to make one for “integer”, I’d want to make sure the back of the card included:

• My own definition in my own words,
• Key trouble issues to watch out for, and
• How the concept tends to show up on the exam.

As I did additional problems, I might add information to the back of the card, so that eventually it would look something like this:

• not decimals or fractions
• Includes zero and negatives!
• When they say “non-negative integer”, think “positive OR zero”
• When they say “number,” think about fractions
• When the exponent is a positive integer, the value usually gets bigger. UNLESS that positive integer is one – value stays the same.

The key is that your definition should include all the things that tripped you up, written in your own language, and written in a way that tells you what to do, not what not to do. (Notice my card doesn’t say anything like, “don’t test only positive numbers”, because generally it’s much harder for us to remember directions given in the negative.) It’s less of a definition and more of a collection of key points that help you clarify how this topic is applied on the exam. In this way, you become a better issue-spotter and avoid common mistakes.

Thinking of definitions in this way can help you to realize their importance while also learning them in a way that’s directly applicable to the exam. The next paragraph is a big, long list of terms for which you might find a definition card useful. All these terms are covered in ETS’s math review for the GRE. You certainly don’t need to make definition cards for each of these words, but if you think it would help you, go for it!

You might find it helpful to make definition cards for the following terms: integer, even, odd, positive, negative, divisible, factor, multiple, greatest common factor, least common multiple, remainder, prime number, prime factor, composite number, zero, one, rational number, reciprocal, square root, terminating decimal, real number, less than, greater than, absolute value, ratio, proportion, percent, percent increase, percent decrease, domain, compound interest, slope, y-intercept, reflection, symmetric, x-intercept, parallel, perpendicular, line of symmetry, parabola, vertex, circle, stretched, shrunk, shifted, line segment, congruent, midpoint, bisect, perpendicular bisector, opposite angles, verticle angles, right angle, acute, obtuse, polygon, triangle, quadrilateral, pentagon, hexagon, octagon, regular polygon, perimeter, area, equilateral triangle, right triangle, hypotenuse, legs, square, rectangle, parallelogram, trapezoid, chord, circumference, radius, diameter, arc, measure of an arc, length of an arc, sector, tangent, point of tangency, inscribed, circumscribed, rectangular solid, face, cube, volume, surface area, circular cylinder, lateral surface, axis, right circular cylinder, frequency, count, frequency distribution, relative frequency, relative frequency distribution, univariate, bivariate, central tendency, mean, median, mode, weighted mean, quartiles, percentiles, dispersion, range, outliers, interquartile range, standard deviation, sample standard deviation, population standard deviation, standardization, finite set, infinite set, nonempty set, empty set, subset, list, intersection, union, disjoint, mutually exclusive, universal set, factorial, probability, permutation, combination, and normal distribution.

To see this week’s answer choices and to submit your pick, visit our Challenge Problem page.

We hope everyone had a happy Halloween! Yesterday we asked our friends on our Manhattan GRE Facebook page to attempt this Trick-or-Treat Halloween Challenge Problem. As promised, today we are sharing the answer and explanation to the problem:

Let’s use x for the number of bags produced by the original recipe, and y for the weight of each of the bags. Given those variables, our first equation is simply xy = 600. We also need to create an equation that represents the new recipe. Since the number of bags produced has increased by 30, and the weight of each bag has decreased by 1, the new equation is (x + 30)(y – 1) = 600. Remember, the total weight is still 600 ounces. Foiling this equation yields xy – x + 30y – 30 = 600.

We now have two equations with two variables. There are several different paths we can go down here, but all involve substitution of one of the variables, and all will yield a quadratic. The simplest path is to recognize that since xy = 600, we can substitute for xy in the second equation to get 600 – x + 30y – 30 = 600. Subtracting the 600 from both sides, and adding an x to each side gives us 30y – 30 = x. We can now substitute for x in the first equation.

 
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Each week, we post a new GRE Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for two free Manhattan Prep GRE Strategy Guides.

Submit your pick over on our Challenge Problem page.

Each week, we post a new GRE Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for two free Manhattan Prep GRE Strategy Guides.

See the answer choices and submit your pick over on our Challenge Problem page.

Each week, we post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for two free Manhattan Prep GRE Strategy Guides.