query: Every positive real number has a unique positive $k$-th root
positive_passages.docid: 16660
positive_passages.text: Prove that any positive number has an $n$th root
negative_passages.docid: 124472
negative_passages.text: Elements with infinite roots in $p$-adic integers
negative_passages.docid: 56818
negative_passages.text: Prove that if $A$ is positive, then $\exists\;\alpha>0$ such that $\langle Ax, x \rangle\geq \alpha|x|^2$ for $x\in\Bbb{R}^n$
negative_passages.docid: 81381
negative_passages.text: Why does the domain and range of $\sqrt x$ contain only positive real numbers?
negative_passages.docid: 78294
negative_passages.text: How does one denote the set of  all positive real numbers?
negative_passages.docid: 16235
negative_passages.text: Prove by induction that every positive integer is represented by some binary number?
negative_passages.docid: 146439
negative_passages.text: Is every sufficiently large positive integer of the form $ab + ac + bc + 1$?
negative_passages.docid: 25981
negative_passages.text: Prove that $\sum_{k=0}^n a_k x^k = 0$ has at least $1$ real root if $\sum_{k=0}^n \frac{a_k}{k+1} = 0$
negative_passages.docid: 127468
negative_passages.text: Prove that there is only one unique base b representation of any natural number.
negative_passages.docid: 13656
negative_passages.text: Prove existence of a real root.
negative_passages.docid: 103014
negative_passages.text: Is the real number structure unique?
negative_passages.docid: 44799
negative_passages.text: Why is the even root of a number always positive?
negative_passages.docid: 107765
negative_passages.text: Prove that there exists two roots in $[-1,1]$
negative_passages.docid: 20438
negative_passages.text: Prove that every positive integer $n$ has a unique expression of the form: $2^{r}m$ where $r\ge 0$ and $m$ is an odd positive integer
negative_passages.docid: 5287
negative_passages.text: Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$
negative_passages.docid: 63182
negative_passages.text: For every $n \ge 1$ there exist uniquely determined integers $a \gt 0$ and $b \gt 0$ such that $n = a^2b$ where $b$ is square-free.
negative_passages.docid: 114043
negative_passages.text: Prove the existence of a real number satisfying a property