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Even n^\text{th} roots are defined only for non-negative real numbers. When n is even and the radicand is positive, two real roots exist; the positive one is known as the principal root. For example, the number 16 has two square roots. ccc & &sqrt(16)=± 4& & &⇕& &4^2=16 &and& (-4)^2=16 The principal root of 16 is 4. In contrast, when considering odd roots, such as cube roots, there is only one real root, which then is the principal root. sqrt(27)=3 ⇔ 3^3=27 Since odd roots are defined for all real numbers, the principal root can be either negative or positive. Suppose that n is an integer greater than 1 and a is a real number.
| n is even | n is odd | |
|---|---|---|
| a> 0 | Two square roots: one positive and one negative. The principal root is the positive one. | One positive root, which is the principal root. |
| a= 0 | The only root is zero. | The only root is zero. |
| a< 0 | No real roots. | One negative root, which is the principal root. |
When dealing with an even n^(th)root, use the principal (positive) root unless the root is applied to both sides of an equation, in which case both the positive and negative roots may be considered.
