Existence conditions of radicals

Not all radicals exist for any value of the argument: it depends on the index and on the position of the root. We must therefore impose the existence conditions (E.C.).

Careful — Existence conditions (E.C.)

• If the index $n$ is even: the argument must be $\ge 0$ (E.C.: $a\ge 0$).
• If the index $n$ is odd: no E.C. needed.
• If the radical is in a denominator: $\sqrt[n]{a}\neq 0$, that is $a\neq 0$ (and $a\ge 0$ if $n$ even).

For example, the E.C. of $\sqrt{x+1}$ is $x+1\ge 0$, that is $x\ge -1$; whereas the E.C. of $\dfrac{1}{\sqrt{x+1}}$ is $x+1>0$, because the root is in a denominator and cannot vanish.

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Topics: Radicals
Concepts: Existence conditions · Radical
Skills: Reasoning by cases