【GMAT Tip】Square Roots Aren’t So Squarey

Radicals are very commonly tested on the GMAT, and frankly, can be overwhelming to a lot of students because there are so many rules to remember. Many students come to us saying, “Wait, I’m so confused!” often overcomplicating the need-to-know items on how to effectively tackle (most) GMAT radical questions. The reality is that many of the properties you know and feel comfortable with surrounding multiplication and division of integers (e.g. even x odd = odd) and exponents are represented with radicals, but in the reverse direction.

But, we know radicals can seem pretty scary! So, we wanted to share a quick cheat sheet of the top things to know in hopes you don’t feel quite so inundated.

The top thing to know about radicals:

When you square (or even fourth or sixth root) root anything, there is always the opportunity that one of the resulting roots is negative.

For example, the square root of 4 can be equal to +2 or -2. Why? Because when we do the reverse operation of raising it to the square power, multiplying a negative by a negative gives us a positive number.

This is the case for any even number root or exponent. But, it doesn’t apply for odd roots – the cubed root of 27 is equal to +3 only, because -3 * -3 *-3 is -27, and the root of a negative number is imaginary.

Top tip number two – adding radicals

Just as when you add exponents the bases have to be the same, so the same rule applies for radicals. For example, you know that 2^2 + 4^2 is NOT equal to 6^2, but instead is 2^2 + (2^2)^2 = 2^4 + 2^4 = 2^6.

Apply the same logic to radicals. 2(square root 3) + 2 (square root 3) is equal to 4 (square root 3) – we just add the numbers in front of the radical. However, 2 (square root 2) + 2(square root 8) must be simplified to 2 (square root 2) + 2 (square root 4 * 2) which again simplifies to 2 (square root 2) + 4 (square root 2) to get to the correct answer of 6 (square root 2).

Top tip number three – fractional exponents

Still feel like you cannot handle radicals because they look really wacky? Understanding that radicals are really just fractional exponents could help you better answer questions.

For example, the square root of 4 is just 4^½ which equals (2^2)^½ which, when the exponent is multiplied, reduces to 2^1 (or 2).

The cubed root of 3 is 3^⅓ and the fourth root of 2 is 2^¼, and so on.

Final tip – forget the rules, just estimate

Often, it can be easier for us to just simply estimate out radicals and roots, particularly when answers are fairly far apart (and when we are adding difficult, messy radicals!).

For example, the square root of 24 will be a little under 5, as the square root of 25 is 5. The square root of 3 will be a little under 2, the square root of 17 a little over 4, and so on. But, the key is to make sure you still know your list of perfect squares!

Hopefully, these top four tips will help you feel a bit more confident when it comes to radicals and roots. In our next post, we will work through some particularly sneaky GMAT questions using these top four tips. Stay tuned!