Common Math Errors on the GRE

by Chelsey Cooley Dec 4, 2019

Quick: here’s an expression from a GRE math problem. How do you simplify it?

√2¹⁴+2¹⁷ 

As a long-time GRE teacher, I like collecting problems like these: ones that often reveal the math misconceptions we don’t even know we have! There are a lot of different wrong ways to simplify this expression. Try it out before you keep reading—then we’ll look at some other math myths and common mistakes, and how to avoid second-guessing on the GRE. 

1. Adding and Subtracting Exponents

First of all, if you added the exponents together, you’re going down the wrong track. Here are some things you can’t do with exponents, and why. 

• If you’re adding two exponents together, you can’t simplify like this: 

a² + b² does NOT equal (a + b)² !

• You also can’t go the other way: 

(a + b)² does NOT equal a² + b²!

• It doesn’t work with subtraction, either. 

a² – b² does NOT equal (a – b)²!

(a – b)² does NOT equal a² – b²!

Double-check this rule using some small numbers. Remember the Pythagorean triples: sets of three integers that can be the side lengths of a right triangle, like 3, 4, and 5. Let a and b equal 3 and 4: 

3² + 4² = 5² = 25

(3 + 4)² = 7² = 49

So, you can’t freely go back and forth between 3² + 4², and (3 + 4)²; they have different values. 

That’s not exactly the situation in the problem from earlier, though. In that case, the two bases are the same, but the two exponents are different: 

2¹⁴ + 2¹⁷

Unfortunately, there’s not a simple rule for adding exponents with the same base, like there is for multiplying or dividing. The only way to actually simplify something like this is to factor out a like term. 

The number 2¹⁴ divides into both 2¹⁴ and 2¹⁷. You can rewrite 2¹⁴ as (2¹⁴)(1), and you can rewrite 2¹⁷ as (2¹⁴)(23). Here’s how to start simplifying: 

√2¹⁴+2¹⁷

√(2¹⁴)(1) +(2¹⁴)(2³)

√(2¹⁴)(1+2³)

√(2¹⁴)(9)

Finally, there’s no more addition, so you can safely take the square root. The square root of 2¹⁴ is 2⁷, and the square root of 9 is 3. So, the answer to the original problem is (2⁷)(3). 

In short: 

• Don’t “split” or “join” bases when you add or subtract exponents. Remember the example of 3² + 4²: it definitely doesn’t equal 7²!
• If the bases are the same, there is something you can do, although it takes a little more work: factor out a common term, then simplify from there. 

By the way, the same goes for square roots. Here’s something else you can’t do with the expression above: 

√2¹⁴+2¹⁷ does NOT equal √2¹⁴+√2¹⁷!

Use the same example, with 3, 4 and 5, to double-check. Because 3, 4, and 5 form a Pythagorean triple, √3²+4² =5. That’s not the same as √3²+√4², which equals 7. 

2. Weird Exponents

When you see something unusual in an exponent—a fraction, a negative number, or a variable—the ultimate math mistake is to panic and bail out. The normal exponent rules still work in the normal way, even when the exponent looks strange. Apply the same rules that you would in an easier problem.  

For example, the rules of exponents say that when you raise an exponent to another power, like (2⁵)⁷, you multiply the two exponents together, getting 2³⁵. 

Okay, what about (2^8x)^-0.5? 

Simplify it in exactly the same way as the previous problem: multiply the two exponents together, giving you 2^-4x. 

Normally, when you divide one exponent by another with the same base, you subtract the exponents from each other, like this: 

2¹⁰/2⁴=2^10-4=2⁶

Use the same process when the exponents look awkward: 

2^7x/4/2^-x/4

2^{(7x/4 – (-x/4))}

2^8x/4

2^2x

Don’t let exponent problems intimidate you into making math errors.

3. Comparing Values

One GRE math problem type is Quantitative Comparisons. Solving these involves comparing two values, so of course the GRE loves to take advantage of a common mistake people make when comparing numbers. 

When you’re comparing two negative numbers, the one that looks larger is actually “smaller”: for instance, -8 is smaller than -5. This might seem obvious, but on test day, it’s easy to let stress get the better of you and make a quick but incorrect assumption. To stay safe, visualize a number line when you compare negative values. The value to the left is the smaller one.

This is particularly useful when comparing negative fractions, which are doubly difficult because of how counterintuitive fractions can be. Ask yourself: which of these negative numbers is further away from zero, or further to the left? That’s the smaller one. 

Use a number line to avoid missing a GRE Quantitative Comparison problem at the last moment!

4. PEMDAS

First, check out this article, which will teach you more about PEMDAS (or BODMAS) than you ever wanted to know. 

If you’re simplifying a GRE math expression using PEMDAS, here’s a common pitfall to avoid: when you’re doing multiplication or division, work from left to right! Contrary to the acronym, you DON’T actually do all of the multiplication first, then all of the division. Instead, you do them both at the same time, in left-to-right order.

So, here’s an expression: 

10 / 5 * 6

And here’s the right way to simplify it: 

2 * 6

12

Here’s what you shouldn’t do: 

10 / 30

⅓

The same is true for addition and subtraction: do them both together, working from left to right. This could save you from some silly math mistakes on the GRE. 

5. Negatives and Parentheses

These two values look similar but are actually different: 

(-3)²

-3²

The first is equal to positive 9, while the second equals -9. That’s actually because of PEMDAS again! Making something negative is the same as multiplying it by -1, so the negation falls under the ‘M’ step of PEMDAS. To simplify the first expression, whatever’s inside of the parentheses comes first: take the number 3, then make it negative. Then, move on to the exponent: take that negative number and square it, making it positive again. 

To simplify the second expression, since there are no parentheses, start with the exponent: 3² equals 9. Then, make it negative, giving you negative 9. 

6. Ratios

You may have noticed a common feature in the answer choices of GRE ratio problems: many of the answers will be nearly identical to each other, but in reverse. For instance, you might see a set of answer choices like these: 

(A) 4 to 7

(B) 3 to 5

(C) 1 to 1

(D) 5 to 3

(E) 7 to 4

That’s the GRE taking advantage of a common math mistake: accidentally flipping a ratio backwards. This falls under the “pure careless mistake” category! Even if you understand ratios perfectly, you might fall for this one. To avoid it, try two things. First, in most ratio problems, the best way to organize your scratch paper is with a well-labeled table or chart. Don’t just jot down the ratios by themselves. Second, a useful “sanity check” is to look back at the problem and decide which part of the ratio should, logically, be bigger. If the problem deals with the number of apples and pears in a fruit salad, and it’s clear that there are more apples than pears, you shouldn’t have a ratio whose first part is smaller than its second part. If you do, you may have flipped it.

Know Your Math Mistakes

When I first started studying for the GRE myself, I remember being tempted to put away a missed problem as soon as I figured out why I’d gotten it wrong. This was especially true if I felt like I “should have” gotten that one right. Of course, you shouldn’t dwell on and beat yourself up over math errors. But there’s a difference between beating yourself up and gathering good data. Really strive to know your errors: identify them, understand them, know where they came from, write them down, and make a real plan to avoid making them a second time. 

Looking for more common math mistakes to study? You can also check out this article from our GMAT blog! The math content on the GRE and the GMAT are very similar, so all of the notes in both articles will apply to both tests. 

Don’t forget that you can attend the first session of any of our online or in-person GRE courses absolutely free. We’re not kidding! Check out our upcoming courses here.

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Chelsey Cooley is a Manhattan Prep instructor based in Seattle, Washington. Chelsey always followed her heart when it came to her education. Luckily, her heart led her straight to the perfect background for GMAT and GRE teaching: she has undergraduate degrees in mathematics and history, a master’s degree in linguistics, a 790 on the GMAT, and a perfect 170Q/170V on the GRE. Check out Chelsey’s upcoming GRE prep offerings here.