Let \(\mathbb R_{+}:=\{x\in\mathbb R, x\ge 0\}\) be the subset of real numbers consisting of positive real numbers, including \(0\). For an integer \(n\ge 2\) or \(n\le -2\), the function \(f:\mathbb R_{ + }\to\mathbb R\), denoted by \(x\to \sqrt[n]{x}\), is continuous and strictly monotonically increasing, and called the \(n\)-th root of the positive number \(x\).
The \(n\)-th root is the inverse function to the \(n\)-th power, i.e. for \(n\ge 2\) or \(n\le -2\), we have
\[(\sqrt[n]{x})^{n}=x\]
and \[(\sqrt[-n]{x})^{-n}=\left(\frac {1}{\sqrt[n]{x}}\right)^{-n}=\left(\frac {\sqrt[n]{x}}{1}\right)^n=x\]
The following interactive figure shows the \(n\)-th root (red) and the \(-n\)-nt root (blue) for \(2\le|n|\le20\):
Proofs: 1
Corollaries: 1 2
Definitions: 3
Proofs: 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46
Propositions: 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77