# GRE Arithmetic - Roots and Square Roots

The concept of **roots** is closely related to the topic of **exponents**, and often appears on the medium and difficult GRE math questions. The formal definition of a **root** is the following:

The *n*-th root of a number *x* is a number *r* which, when raised to the power of *n*, equals to *x*

In other words: *r ^{n} = x*

For example, 2 is a 4th root of 16 because 2^{4} = 16. The number *n* is called the degree of the root. A root of degree 2 is called a **square root**, a root of degree 3 is called a **cube root**, a root of degree 4 is called a **fourth root**, and so on. In general, a root of degree *n* is called an ** nth root**. The square root of a number, because it is a concept that is used often in various disciplines, has its own special symbol:

Square root of 9 = √9 = 3, because 3×3=9

Cubed roots (and, in fact any n-th root, but we'll spare you the details) can also be written using a symbol notation.

Cubed root of 8 = ∛8 = 2, because 2×2×2=8

So how does this all relate to exponents? Well, roots are treated as special cases of exponentiation, where the exponent is a fraction:

Square root of 9 = √9 = 9^{1/2} = 3

Cubed root of 8 = ∛8 = 8^{1/3} = 2

The topic of roots and square roots could be included in the Algebra section of the tutorials, but we are including it here because, often-times, when you are asked a math question on the GRE exam that attempts to measure your knowledge of roots, the question will not include variables such as *x* or *r*, but instead will contain only numbers.