n th Roots

A square root of a number b , written b , is a solution of the equation x 2 =b .

Example: 49 =7 , because 7 2 =49 .

Similarly, the cube root of a number b , written b 3 , is a solution to the equation x 3 =b .

Example: 64 3 =4 , because 4 3 =64 .

More generally, the n th root of b , written b n , is a number x which satisfies x n =b .

The n th root can also be written as a fractional exponent:

b n = b 1 n

When does the n th 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 n th root of b exists whenever b is positive; and  for all b .
  • If n is an odd whole number, the n th root of b exists for all b

Examples:

81 4 is not a real number.

32 5 =2

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:

3 4 =81 ( 3 ) 4 =81 ( 3i ) 4 = 3 4 i 4 =81 ( 3i ) 4 = ( 3 ) 4 i 4 =81