nth Roots

A square root of a number b, written , is a solution of the equation .

Example: , because .

Similarly, the cube root of a number b, written , is a solution to the equation .

Example: , because .

More generally, the nth root of b, written , is a number x which satisifies .

The nth root can also be written as a fractional exponent:

When does the nth root exist, and how many are there?

If you are working in the real number system only, then

  • If n is an even whole number, the nth root of b exists whenever b is positive
  • If n is an odd whole number, the nth root of b exists for all b

Examples:

If you are working in the complex number system, then things get more, well, complex.

Here every number has 2 square roots, 3 cube roots, 4 fourth roots, 5 fifth roots, etc.

For example, the 4 fourth roots of the number 81 are 3, −3, 3i, and −3i. Because: