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[parent] existence of $n$th root (Theorem)
Theorem   If % latex2html id marker 201 $ 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 % latex2html id marker 220 $ 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 \neq 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. $ \qedsymbol$



"existence of $n$th root" is owned by Wkbj79. [ full author list (2) | owner history (1) ]
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Cross-references: interval, strictly increasing, function, intermediate value theorem, continuous, polynomial function, root, real number, integer, positive
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This is version 18 of existence of $n$th root, born on 2006-04-25, modified 2007-04-21.
Object id is 7867, canonical name is ExistenceOfRoot.
Accessed 1313 times total.

Classification:
AMS MSC: 12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)
 26A06 (Real functions :: Functions of one variable :: One-variable calculus)
 26C10 (Real functions :: Polynomials, rational functions :: Polynomials: location of zeros)
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