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Exponents

nth Root

A real number r is called an nth root of b if rn = b.

The 2nd root is also called the square root and the 3rd root is called the cube root.

In the following we assume b is a positive number, and n a positive integer.

Notation:

b^(1/n) (or the nth root of b ) is defined to be the positive nth root of b. Thus (b^(1/n))^n = b .

b^(1/2)=sqrt(b) .

b^(1/3) = cube root of b.

 

If n is even:

  • As bn > 0 for all real numbers b, negative numbers have no real nth root.
  • Any positive number b has two nth roots.
  • The number 0 has only one nth root, namely 0.

Example

24 = 16 and  (-2)4 = 16,  so 161/4 = 2 and 161/4 = −2.

If n is odd:

  • b1/n is positive for all positive numbers b.
  • b1/n is negative for all negative numbers b.
  • The number 0 has only one nth root, namely 0.

Examples

105 = 100000,  so 1000001/5 = 10.

(-10)5 = −100000,  so (−100000)1/5 = -10.

Have a look at a few more examples. Any decimal roots are approximate only.  None means the number does not have an nth root.

b
n
b1/n

 

Exercise

In the following exercise, if the nth root exists and is an integer display it in the box provided. If it exists but is not an integer, type NI for "not integer". Otherwise, if the root does not exist, type N for "no solution".

b n b1/n

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