---
tag: 0J29GM30Z7
kind: example
name: Existence of nth roots by the intermediate value theorem
msc: [26A15]
uses: [00KAFBR32E, 0C5TP6RYSV]
sources:
  - url: https://en.wikipedia.org/wiki/Nth_root
    locator: Definition and notation
---

## Example

Let $n \ge 1$ be a natural number and let $y \ge 0$ be real.
Then there is a unique real $x \ge 0$ with $x^n = y$.
For existence, let $b = \max(1, y)$. The function $p(x) = x^n$ is [[continuous|0C5TP6RYSV]], with $p(0) = 0 \le y$ and $p(b) \ge b \ge y$.
The [[intermediate value theorem|00KAFBR32E]] gives $x \in [0, b]$ with $x^n = y$.

## Remarks

Uniqueness holds because $x \mapsto x^n$ is strictly increasing on $[0, \infty)$.
The number $x$ is the $n$-th root $\sqrt[n]{y}$.
For example, it proves that $\sqrt{2}$ exists as a real number.

## Explanations

### L1 human

The graph of $x^n$ starts at $0$ and grows past $y$.
Since the graph has no gaps, it reaches the height $y$ at some point, which is the root.