One of the most common mistakes that I see students make when practicing for the GMAT is the misapplication of the rules that govern square roots. When approaching a question that involves radicals, it is vital that you know not only the rules that you must follow, but also the operations that are commonly believed to be rules, but are not. On test day, the wrong answer choices will almost always be derived from the latter.

If you need to manipulate a square root, you must remember two key rules. First, that √(ab) = √a x √b and, second, that the √(a/b) = √a/√b. For example, if you need to simplify √20, you can rewrite it as √(4×5). When choosing which factors to use, always look for perfect squares. Since 4×5 includes the perfect square 4, it is better than 2×10, which does not include a perfect square. Next, you can break this down to √4√5, which in turn becomes 2√5. You can follow the same steps in division problems.

However, you must also remember that √(a+b) ≠ √a + √b and that √(a-b) ≠ √a – √b. To understand why you cannot break apart a radical across an addition or subtraction sign, consider the expression √(4+9). We could add the four and nine together, giving us √13, but since 13 is not a perfect square and is also prime, we are unable to break the expression down any further. If we mistakenly believed we could break apart the radical around the addition sign, we would end up with √4 + √9 = 2 + 3 = 5. Since the square root of 13 does not equal 5, we know that this operation is not allowed. If we tried a similar exercise with subtraction, we would find a similar outcome.

Keep in mind when you can and cannot break apart a radical as you give the problem below a try:

**Problem:**

**Solution:**

*Analyze*

What do we see in this Q-stem? A complex set of operations under a radical: multiplication, division, and addition.

*Task*

We’re just simplifying the original expression.

*Approach Strategically*

Take this piece-by-piece. First, separate the numerator and denominator under their own radicals:

Now, the denominator is easily simplified. It’s a perfect square:

Next we can start simplifying the numerator. We can factor the 9s out of the expression to get another perfect square:

The square root of 9 is 3, so we can put that outside the radical:

It’s now clear that (C) is incorrect. To get one step closer, factor 20 into 4 x 5. Since 4 is a perfect square, we can get a little more out from under the radical:

The answer is **(E).**

**Are you all clear on this one? Use the comments to let me know where you’re getting stuck, and I can help.**

Bret has been teaching for Kaplan since 2005, and has helped over 1000 students with their GMAT preparation. He spent three years teaching in Manhattan, where he served as an Elite Teacher and a full-time instructor, before moving to London, where he is now the GMAT Master Teacher for Kaplan’s London Center. As the GMAT Master Teacher, Bret trains, observes and mentors teachers, in addition to continuing his own teaching and tutoring, and has taught courses across Europe, including Italy, Ireland, and Germany. Bret contributes to Kaplan’s GMAT curriculum on an on-going basis, and was also a contributor to Kaplan's 2010 GMAT course.