- #1
silimay
- 26
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Homework Statement
Let n and k be positive integers. Show that [tex]k^{1/n}[/tex] is either a positive integer or an irrational number.
The Attempt at a Solution
I set [tex]q = k^{1/n}[/tex]. Then I set [tex]q = \frac{m}{p} [/tex]. (Where m and p don't have common factors.) Then [tex]m^n = k * p^n [/tex]. So then k is a factor of [tex]m^n[/tex].
But here I get stuck. In other proofs they usually show that, like, then k must also be a factor of m, (but I don't know how to do that, if it is true), and then so [tex]m^n = [/tex] an integer * k^2, so then k must also be a factor of [tex]p^n[/tex], which means that m and p do have a common factor.
But I get stuck in the middle.
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