GMAT Exam: A Simple Guide To Arithmetic - Roots and Radicals

Square Roots
Every positive number has two square roots that are equal in absolute value, but opposite in sign.

Zero has only one square root, namely 0.

For instance, 4 and –4 are the two square roots of 16.

The positive square root is the principal square root.

The square root radical always denotes the principal square root.

Thus, and . Tip: , not –4 or ±4.

You indicate the negative square root of 16 as . Thus, .

To be well-prepared for the GMAT, you should memorize the following 20 principal square roots:


On the GMAT, don’t try to find square roots of negative numbers, because no real number will multiply by itself to give a negative number.

Cube Roots
Every real number has one real cube root, called its principal cube root.

For example, because 4 × 4 × 4 = 64, 4 is the principal cube root of 64. Likewise, because (–4)(–4)(–4) = –64, –4 is the principal cube root of –64.

As you can see, the principal cube root of a positive number is positive, and the principal cube root of a negative number is negative.

You use the cube root radical to designate the principal cube root.

The small number 3 in the radical indicates that the cube root is desired.

This number is the index of the radical. Thus, and .

Here is a list of principal cube roots of some positive perfect cubes.


For the GMAT, it would be a good idea for you to know these roots.

Higher Roots
In general, the index of the radical tells you which root to find.

Tip: The index for the square root radical is understood to be 2.

Here are examples of higher roots.


The number under the radical is the radicand. Here are some points to keep in mind:
- If the index is even and the radicand is positive, the principal root is positive.
- If the index is even and the radicand is negative, there is no real root.
- If the index is odd and the radicand is positive, the principal root is positive.
- If the index is odd and the radicand is negative, the principal root is negative.