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: rn = x

For example, 2 is a 4th root of 16 because 24 = 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 = 91/2 = 3
Cubed root of 8 = ∛8 = 81/3 = 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.