Simplifying Radical Expressions on the GMAT

Let’s start out with a couple practice questions. While these look like geometry questions, you’ll have to put your GMAT algebra skills to work!

Practice questions

2) What is the length of the diagonal of a square with area 48?

The geometry of those questions is not too difficult, but the trick is how to handle those radicals. If that befuddles you, this is just the post for you.


Distributing radicals

First of all, we need to review some rules for square-roots. We can distribute a square-root over multiplication or division. We cannot distribute a square-root over addition or subtraction. Another way to say that: we can separate a square-root by multiplication or division, but not by addition or subtraction.

Beware of those two on the right: they are very tempting mistakes, and GMAT questions will be designed specifically to elicit that mistake.

Improve your GMAT score with Magoosh.


Simplifying radicals

Suppose we want to simplify \(sqrt(72)\), which means writing it as a product of some positive integer and some much smaller root. In order to do this, we are going to use the first property given in the previous section: we can separate the square-root by multiplication. The trick is: for 72, find any factor which is a perfect square. For example, \(72 = 9*8\), and 9 is a perfect square. Therefore

We have simplified, but not all the way, because 8 still has another factor, 4, which is a perfect square. To continue:

That last expression is fully simplified, because the number under the radical, 2, has no factor greater than 1 that is a perfect square. (You always are guaranteed the radical is as simple as it’s going to get if the number under the radical is prime.) That simplification took a relatively large number of steps. We would have been more efficient if we had noticed at the outset: \(72 = 36*2\). Then:

It saves you a bit of time if you happen to factor out the largest possible perfect-square factor right at the beginning, but as this process shows, that’s not strictly necessary for arriving at the correct simplification.

Having seen this, you may want to give the practice problems another go before looking at the explanations below.


Practice problems solutions

1) First of all, we use that most amazing theorem, the Pythagorean Theorem.

That’s the easy part. Now, we have to take a square-root of 80. Notice that 80 = 16*5 (16 is the largest perfect-square factor of 80).

Answer = B

2) There are a few dozen ways to approach this. Consider the square:

Improve your GMAT score with Magoosh.

We know that \(s^2 = 48\). Let’s not simplify that, but just leave that as is. You see, we are going to use the Pythagorean Theorem again, in the right triangle ACD, so even if we found out the square-root of s, we would have to turn around and square it again anyway.

AAA Pythagorean correction

Again, the meat of the problem involves simplifying this radical. Notice that \(96 = 16*6\).

Answer = C


Ready to get an awesome GMAT score? Start here.

Most Popular Resources


  • Mike MᶜGarry

    Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as "member of the month" for over two years at GMAT Club. Mike holds an A.B. in Physics (graduating magna cum laude) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations and resources like What is a Good GMAT Score? and the GMAT Diagnostic Test.

More from Magoosh