existence of $n$ th root

Theorem.

If $a\\in\\mathbb{R}$ with $a>0$ and $n$ is a positive integer, then there exists a unique positive real number $u$ such that $u^{n}=a$ .

Proof.

The statement is clearly true for $n=1$ (let $u=a$ ). Thus, it will be assumed that $n>1$ .

Define $p\\colon\\mathbb{R}\\to\\mathbb{R}$ by $p(x)=x^{n}-a$ . Note that a positive real root of $p(x)$ corresponds to a positive real number $u$ such that $u^{n}=a$ .

If $a=1$ , then $p(1)=1^{n}-1=0$ , in which case the existence of $u$ has been established.

Note that $p(x)$ is a polynomial function and thus is continuous. If $a<1$ , then $p(1)=1^{n}-a>1-1=0$ . If $a>1$ , then $p(a)=a^{n}-a=a(a^{n-1}-1)>0$ . Note also that $p(0)=0^{n}-a=-a<0$ . Thus, if $a\eq 1$ , then the intermediate value theorem can be applied to yield the existence of $u$ .

For uniqueness, note that the function $p(x)$ is strictly increasing on the interval $(0,\\infty)$ . It follows that $u$ as described in the statement of the theorem exists uniquely.∎

单词	ExistenceOfNthRoot
释义	existence of $n$ th root Theorem. If $a\\in\\mathbb{R}$ with $a>0$ and $n$ is a positive integer, then there exists a unique positive real number $u$ such that $u^{n}=a$ . Proof. The statement is clearly true for $n=1$ (let $u=a$ ). Thus, it will be assumed that $n>1$ . Define $p\\colon\\mathbb{R}\\to\\mathbb{R}$ by $p(x)=x^{n}-a$ . Note that a positive real root of $p(x)$ corresponds to a positive real number $u$ such that $u^{n}=a$ . If $a=1$ , then $p(1)=1^{n}-1=0$ , in which case the existence of $u$ has been established. Note that $p(x)$ is a polynomial function and thus is continuous. If $a<1$ , then $p(1)=1^{n}-a>1-1=0$ . If $a>1$ , then $p(a)=a^{n}-a=a(a^{n-1}-1)>0$ . Note also that $p(0)=0^{n}-a=-a<0$ . Thus, if $a\eq 1$ , then the intermediate value theorem can be applied to yield the existence of $u$ . For uniqueness, note that the function $p(x)$ is strictly increasing on the interval $(0,\\infty)$ . It follows that $u$ as described in the statement of the theorem exists uniquely.∎
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existence of nth root

Theorem.

Proof.

existence of $n$ th root