# Proposition: Nth Roots of Positive Numbers

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 $(\sqrt[-n]{x})^{-n}=\left(\frac {1}{\sqrt[n]{x}}\right)^{-n}=\left(\frac {\sqrt[n]{x}}{1}\right)^n=x$

### Notes

• For $$n=2$$, we call $$\sqrt x$$ the square root of $$x\ge 0$$.
• $\sqrt[n]x$ can also be written as the generalized power of $x$, i.e. as $$\sqrt[n]x:=x^{\frac 1n}=\exp_x\left(\frac 1n\right).$$
• The nth root is defined for $x > 0$ and is itself always positive, since the exponential function is always positive.

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

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