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Principal Root

Even roots are defined only for non-negative real numbers. When is even and the radicand is positive, two real roots exist; the positive one is known as the principal root. For example, the number has two square roots.
The principal root of is In contrast, when considering odd roots, such as cube roots, there is only one real root, which then is the principal root. Since odd roots are defined for all real numbers, the principal root can be either negative or positive. Suppose that is an integer greater than and is a real number.
is even is odd
Two square roots: one positive and one negative. The principal root is the positive one. One positive root, which is the principal root.
The only root is zero. The only root is zero.
No real roots. One negative root, which is the principal root.
When dealing with an even 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.


Why the Principal Root Exists

The principal root is an important concept that guarantees functions involving radicals are well-defined. For example, when defining the square root function as it is implied that returns the principal square root of This is to guarantee that the function has a unique output for each input value.

Graph of the square root function
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