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Exponents

n^th Root

A real number r is called an n^th root of b if rⁿ = b.

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

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

Notation:

(or ) is defined to be the positive n^th root of b. Thus .

.

.

If n is even:

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

Example

2⁴ = 16 and  (-2)⁴ = 16,  so 16^1/4 = 2 and 16^1/4 = −2.

If n is odd:

• b^1/n is positive for all positive numbers b.
• b^1/n is negative for all negative numbers b.
• The number 0 has only one n^th root, namely 0.

Examples

10⁵ = 100000,  so 100000^1/5 = 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 n^th root.

Exercise

In the following exercise, if the n^th 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".

<< Cube Root | Exponents Index | General Fractional Exponents >>